Math. Z. 222, 97- 134 (1996)
Mathematische
Zeltschrift
© Springer-Verlag 1996
Nonpositively curved metrics on 2-polyhedra
W. BaHmann,S. Buyalo*
Universit~t Bonn, Mathematisches Institut, Wegeler Strasse 10, D-53115 Bonn, Germany
Received 16 March 1994; in final form 6 September 1994
It follows from the Gauss-Bonnet formula that any Riemannian metric on the
2-torus with nonpositive Gaussian curvature is fiat. Moreover, the space o f fiat
metrics on the torus can be described explicitly. We are looking for similar
results for more general kinds o f metrics on more general kinds of spaces. Our
research was motivated by the question of Gromov whether a word hyperbolic
group admits a cocompact and properly discontinuous action on some space
with a metric o f negative curvature, cf. [G2]. We will not provide an answer
to this question. We obtain, however, some results in this direction; namely
non-existence o f F-invariant metrics o f negative curvature on certain spaces X
with a given cocompact and properly discontinuous action o f a group F, even
if F is hyperbolic. To a large degree our arguments are variations or extensions
o f the above argument for the 2-torus.
Recall that a metric d on a topological space X has curvature at most K
if distances in X are realized by geodesics and if sufficiently small geodesic
triangles in X are thinner than in the model surface of constant curvature
K. The metric d associated to a Riemannian metric on a given manifold has
curvature at most K in this sense if and only if the sectional curvature is at
most K. However, the above concept is much more general and is, in particular,
invariant under more general limits. It goes back to A.D. Alexandrov and, in
the case K = 0, also to Busemarm; cf. [A1, AZ].
There is a large and interesting class o f polyhedra which admit metrics o f
nonpositive curvature. N o w for a given finite 2-polyhedron Y, we are interested
in the space J - ( Y ) o f equivalence classes o f such metrics, where two metrics
are called equivalent if they coincide modulo a homeomorphism isotopic to the
identity. For example, we have the following result, which is a special case o f
Theorem 1.2 and Theorem 5.6 below.
* The first author was supported in part by the EC-project GADGET and the IHES in Buressur-Yvette. The second author was supported by the Sonderforschungsbereich256 at the University
of Bonn.
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w. Ballmann, S. Buyalo
Theorem 1 For n > 3 let Zn be the 2-skeleton of the (n + 1)-simplex and let
P " Yn --~ Zn be the branched covering of Zn of degree 2 with branching locus
the barycenters of the faces of Z,. Then
(i) for any metric of nonpositive curvature on Y,, the faces of Yn are
flat and the edges are geodesics;
(ii) moreover, if n >= 4, then the faces are centrally symmetric flat
hexagons with interior angles 2n/3.
In particular, if e is an edge of Z, and n > 4, then the lengths of the two
edges in Yn above e coincide. Denoting this common length by la(e), we have
(iii) the map e ~ la(e) defines an isomorphism between J-(Y) and IRE+,
where E is the set of edges of Zn.
In the case n = 2, the above space Y, is the 2-torus and Z~ is the 2-sphere.
For all n >__ 2, Y~ is the union of embedded tori, and the proof of assertion
(i) relies on this fact and the fiat torus theorem. Now if e is the edge in the
link Av of the vertex v in Y~ corresponding to the face f adjacent to v, we let
~a(s) be the interior angle of f at v (with respect to d). This defines a labeling
of the edges of Av. The proof of (ii) relies, among others, on the observation
that the constant labeling ~a = 2n/3 is the only possible one for metrics d of
nonpositive curvature on Y,, n > 4.
In combination with the Gauss-Bonnet formula the labelings of the links A,
turn out to be an essential tool. If A is a finite graph representing a link and EA
denotes the set of edges of A, then we are interested in labelings l : EA ~ IR+
such that any simple loop c in A has length at least 2n with respect to l; here,
by definition the length of c is I(c) --- ~ l(e), where the sum is over all edges
contained in c. The condition on the length of simple loops for the labelings
of the links is a necessary condition for a (piecewise smooth) metric of a
2-polyhedron to have curvature bounded from above. For many 2-polyhedra
the condition of nonpositive curvature even implies that the labelings on the
links are as short as possible with respect to the above condition on the length
o f simple loops. These minimal labelings of a given graph constitute a convex
set in the space of all labelings of the graph. If this set consists of one point
only, we say that the graph is metrically rigid.
Theorem 2 Suppose X is a 2-dimensional thick Euclidean Tits building and
F is a properly discontinuous and cocompact group of automorphisms of X.
Then any piecewise smooth F-invariant metric of nonpositive curvature on X
is hornothetic to the standard metric on X.
For example, if p is a prime number and X is the Tits building associated to SI3(~) with respect to the p-adic valuation, see [Bro], then Sl3(@p)
operates by automorphisms on X, transitively on the faces of X. Any discrete
cocompact subgroup F of SI3(Qp) acts properly discontinuously and cocompactly on X since the stabilizers of the vertices of X are compact in Sl3(~p).
Now the existence of discrete cocompact subgroups T of SI3(@p), and even
of torsionless ones, follows from the work of Borel-Harder in [BH].
Metrics on 2-polyhedra
99
Theorem 2 is an application o f Theorem 5.9 below (cf. Example 5.10(a)).
We use that the metric is piecewise smooth when applying the Gauss-Bonnet
formula. This formula implies that all the faces of X are fiat and all the edges
of X are geodesics. The assertion then follows from the fact that the labelings
of the links of the vertices of X have to be minimal and that these links are
metrically rigid (as proved in Sect. 4).
It seems that the higher dimensional version of Theorem 2 also holds. In
fact, as Gromov pointed out to us, the density of immersed fiat tori in a compact
quotient of a Euclidean Tits building holds exactly as in the case of a compact
quotient of a symmetric space of noncompact type, and then arguments as in
Sect. 1 below should prove rigidity.
Our third example shows that there are 2-polyhedra with hyperbolic fundamental group and without piecewise smooth metrics of negative curvature.
Theorem 3 There is a 2-polyhedron Y which is the union of a finite number
of squares such that
(i) the piecewise smooth metric do on Y, such that all the faces of Y
are unit squares, has nonpositive curvature;
(ii) with respect to any piecewise smooth, nonpositively curved metric on
Y, all the faces of Y are flat and all the edges of Y are geodesics;
(iii) the fundamental group of Y is hyperbolic.
The Gauss-Bonnet formula also applies in this example and implies (ii).
The proof of (iii) relies on the result of Eberlein and Gromov that zrI(Y) is
hyperbolic iff the universal cover I7 of Y does not contain an isometrically
embedded totally geodesic Euclidean plane, see [Eb] and [G1, p. 119].
As we said above we assume that the metric is piecewise smooth in order
to be able to apply the Gauss-Bonnet formula. This formula applies to a much
wider class of metrics, see [AZ] and [Re], and, for all we know, to any metric of nonpositive curvature. Related to this there is the following question
concerning the local behavior of nonpositively curved metrics.
Problem. Suppose X is the union of n > 3 squares along a common edge e.
I f d is a nonpositively curved metric on X, which regularity does e inherit?
Is e rectifiable, can e be well approximated by broken geodesics, that is, with
~ i [ i t - ~il uniformly bounded and ai the angles between consecutive pieces?
I f the answer is positive, the words "piecewise smooth" can be omitted in all
our results.
There is a another natural problem related to our investigations, namely the
completion of the space of metrics of nonpositive curvature on a given finite
2-polyhedron or 2-orbihedron. We make a first step in this direction in Sect. 5,
but plan to come back to this at a later discussion.
Structure of the paper. In Sect. 1 we use the flat torus theorem on the one
hand and the existence of a sufficiently large automorphism group on the
other to show that metrics of nonpositive curvature on certain 2-polyhedra
100
w. Ballmann, S. Buyalo
or 2-orbihedra are piecewise flat, that is, all the faces are fiat. In many cases
we can also conclude that all the edges are geodesics.
In Sect.2 we prove the Gauss-Bonnet formula for piecewise smooth metrics
on 2-polyhedra and 2-orbihedra. This formula is of some interest by itself and
its proof is independent of the rest of the paper.
In Sect. 3 we use the Gauss-Bonnet formula to establish conditions which
imply that for a large class of spaces any piecewise smooth nonpositively
curved metric is tight, that is, the faces are fiat, the edges are geodesics and
the induced labelings on the links are minimal.
Motivated by the results in Sect. 3 we then proceed, in Sect. 4, with the
investigation of the space of minimal labelings on a graph. We obtain some
general results and exhibit the space of minimal labelings in some concrete
examples.
In Sect. 5 we study the space ~--(Y). In some concrete examples we determine an explicit parametrization of J-(Y). We also investigate the closure of
f ( Y ) for certain 2-polyhedra Y. In Sect.6 finally we exhibit examples of orbihedra (X, F) with F hyperbolic, but where any nonpositively curved P-invariant
metric on X is tight.
For lack of reference we have included an Appendix where we prove a
useful criterion for a piecewise smooth metric to have an upper curvature
bound (in the sense o f Alexandrov).
Conventions, Definitions and Notations. We say that a 2-dimensional CWcomplex X is a 2-polyhedron if
(.)
the attaching maps of the faces of X are locally injective.
Then the boundary of a face f in X is a loop of edges without U-tums, and
we also refer to f as a polygon or a k-gon, where k = k(f) is the number
of edges in the boundary of f (counted with multiplicity). The barycentric
subdivision of a 2-polyhedron has the stronger property that all the attaching
maps of its faces are homeomorphisms. The second barycentric subdivision is
simplicial. By Vx,Ex and Fx we denote the set of vertices, edges and faces
of X respectively.
For any vertex v in X, the link Av is the graph whose vertices represent the
edges going out of v and whose edges represent the faces of X adjacent to v
according to their multiplicity. That is, if the attaching map of a face f passes
n times through the vertex v, then f gives rise to n edges in A~,, connecting
corresponding vertices of Av.
A 2-orbihedron (X, F) consists of a 2-polyhedron X and a group F which
acts properly discontinuously and by automorphisms on X. A 2-orbihedron
(X,P) is called finite i f X has only finitely many cells modulo F. We say that
a 2-orbihedron (X, F) is closed if it is finite, if X is connected and if every
edge of X is adjacent to at least two faces.
A metric for an orbihedron (X,P) is a F-invariant metric on X. Such a
metric will be called piecewise smooth if X has a F-invariant triangulation
Metrics on 2-polyhedra
101
such that d is induced by smooth Riemannian metrics on the closed simplices (compatible on their intersections). If (X,F) is finite, then the topology induced by a piecewise smooth metric coincides with the usual topology
of X.
Recall that the girth of a graph A is the least possible number of edges
contained in a simple loop of A. By our assumption (.), the girth of the links
Av is a least 2. For a graph A we denote by VA and EA the set of vertices
and edges of A.
1 Rigidity due to flat tori or large automorphism group
There is a class of 2-polyhedra and, more generally, 2-orbihedra on which
every geodesic metric of nonpositive curvature is piecewise flat or even tight.
We indicate two sources for this phenomenon: the flat toms theorem and a
sufficiently large automorphism group. Let us start with a simple example which
shows how the flat torus theorem enforces piecewise flatness.
1.1 Example. Let Y = Ti k-iF T2 be the union of two tori along a common face
F homeomorphic to the disc D 2. Then the restriction of a given nonpositively
curved metric d on Y to 7"1 and T2 is fiat and F is isometric to a closed
convex domain in the Euclidean plane. Indeed, the universal cover X of Y is
the union tAE~ of subspaces homeomorphic to IR2, such that Ei fq Ej is either
empty or a copy of F and such that each Ei is invariant under a subgroup Fi
of el(Y) isomorphic to 772 with Ei/Fi equal to /'1 or T2. Since d i m X = 2,
Ei is the unique embedded plane in X invariant under F i and Fi-equivariantly
homotopic to Ei. Applying the fiat torus theorem we conclude that Ei is fiat
and totally geodesic with respect to the induced geodesic metric d on X. Hence
our claim.
Although the topology of Y limits the possibilities for metrics of nonpositive
curvature very strongly, the space of metrics of nonpositive curvature on Y is
infinite dimensional. We discuss this in more detail in Sect. 5.
This example motivates the following general result.
Fig. 1
102
W. Ballmann, S. Buyalo
1.2 Theorem Let (X, F) be a finite 2-orbihedron with X simply connected.
Assume X is the union UEi of a F-invariant family (Ei)i~t of subspaces
homeomorphic to IR2 such that
(i) there exists an N > 0 such that Ei n Ej, i 4:j, consists of at most N
faces;
(ii) there exists a metric do of nonpositive curvature for (X, F) such that
each Ei, i E I, is isometric to the Euclidean plane.
Then any nonpositively curved metric d for (X, F) is piecewise flat.
Furthermore, if every face F of X is equal to an intersection Ei N Ej, then
the edges of X are geodesics of d.
Proof It is sufficient to prove that each Eg is periodic, i.e., the stabilizer Fi
of Ei acts cocompactly on Ei. Then Fi is a finite extension of 7Z2. Now, as
in (1.1), since dimX = 2, Ei is the unique embedded plane in X invariant
under Fi and Fi-equivariantly homotopic to Ei. Therefore, Ei is flat and totally
geodesic by the flat toms theorem and Ei • Ej is convex with respect to d.
Hence the assertion is reduced to the above claim about stabilizers.
Now endow X with the metric do as in (ii) and argue as in the proof of
Theorem 1.11 in [BB]: X/F is compact and therefore a finite orbihedron. Hence
for any Ei there exists a face f in Ei such that
A = {y E F i r ( f ) C E,}
is infinite and [.J~ea ~;(f) is a net in El. By condition (i) the set {~-l(Ei)[ 7 C A}
is finite (for i fixed). Therefore the stabilizer Fi of Ei contains an infinite set
2; such that {~;(f)[v E S} is a net in Ei. Hence Fi is cocompact on E~. []
1.3 Examples. (a) For n __> 3 let Zn be the 2-skeleton of the (n + 1)-simplex.
There is a (unique) branched covering p : Yn --* Z, of degree 2 with branching
locus the barycenters of the faces of Zn. Then Yn has the same number of faces
as Zn, but each face is hexagonal. The link of every vertex of Yn is isomorphic
to the l-skeleton of the n-simplex. Now Y~ and, more precisely, the universal
covering Xn of Yn together with the group F,, of covering transformations satisfy
the assumptions of Theorem 1.2. To see this take as do the geodesic metric
on Xn such that all faces are regular Euclidean hexagons of side length 1.
Then each face of X, is contained in, and is the intersection of, exactly two
Euclidean planes.
The most remarkable case is n = 3. The cell complex )(3 is the 2-skeleton
of the dual of the tesselation of the hyperbolic space H 3 by ideal regular
3-simplices. In particular, Aut(X3) is a lattice in Iso(H 3) and F3 is the fundamental group of a hyperbolic 3-manifold of finite volume with five ends,
whose spine coincides with Y3.
(b) One can generalize (a) as follows. Let A be a finite graph with the
following properties:
(i) every essential loop of edges consists of at least three edges;
(ii) every edge is contained in at least two essential loops of three edges.
Metrics on 2-po|yhedra
103
Let F = FA be the Coxeter group
r = ( V ; v 2 = 1, (vw) 3 = 1 i f v w E E } ,
where V = VA is the set of vertices of A and E = EA is the set of edges of A.
The Cayley complex X = XA of the above presentation of F is the 2-skeleton
o f the dual of the Coxeter complex of F, and (X, F) satisfies the assumption
of Proposition 1.2: As do we may take the geodesic metric on X such that
all faces are regular Euclidean hexagons of side length 1 (due to the relations
(vw) 3 = 1, vw E E, each face of X has exactly 6 edges). Condition (i) implies
that do has nonpositive curvature. Now if uv * vw * wu is an essential loop
of three edges in A, then the Cayley complex of the subgroup generated by
u, v and w is a totally geodesic Euclidean plane (tesselated by hexagons) in
X. Condition (ii) implies that every face of X is the intersection of at least 2
Euclidean planes containing it.
As concrete examples of graphs satisfying (i) and (ii) one may take the
1-skeletons of the n-simplex for n > 3 (Example (a) for n = 3), of the
n-octahedron for n > 3, or of the icosahedron.
We now discuss the case where the complex X may not contain any fiats
but where any sufficiently homogeneous metric of X of nonpositive curvature
is piecewise fiat.
1.4 Proposition Let k be one of the numbers 3,4 or 6. For (X,F) a f n i t e
orbihedron assume that all the faces of X are k-gons, i.e., have exactly k
edges, and that for each vertex v of X
(i)
the link
(ii)
(iii)
the isotrophy group F~, at v acts transitively on the set of edges of
A~ at v;
each vertex ~ of Av is an isolated fixed point of some 7 E Fv;
the girth of A~ is 2k/(k - 2) = 6,4 or 3 respectively.
Then every metric for (X, F) of nonpositive curvature is piecewise fiat with
geodesic edges.
Proof Let d be a F-invariant metric on X of nonpositive curvature. Condition
(ii) implies that all edges of X are geodesics. Condition (i) implies that all the
angles of faces at v are equal. Since the metric d has an upper bound for the
curvature (namely 0), the complete angle of an essential loop in A~, must be
at least 27z. Hence the angle of the faces at v is at least ( k - 2)zc/k by (iii).
Since the metric has nonpositive curvature and all faces are k-gons with
geodesic edges and interior angles at the vertices at least ( k - 2)Tz/k, all interior
angles are equal to exactly ( k - 2)folk and all faces are flat. []
1.5 Example. According to Theorem 2.7 in [BB] there exist precisely two
simply connected 2-polyhedra X such that
(i) the link Av at any vertex v of X is isomorphic to the 1-skeleton A of
a given Platonic solid;
104
w. Ballmann, S. Buyato
(ii) the faces of X are k-gons, where k > 2 # / ( # - 2) and # is the girth
of A;
(iii) for any two vertices v,w of X and any isomorphism Av ~ Aw there
exists precisely one automorphism of X inducing ¢p.
Hence for k = 2 # / ( # - 2) and F = Ant(X), the assumptions of
Proposition 1.4 are satisfied. O f the two examples, the "untwisted one" satisfies the assumption of Proposition 1.2 as well. The other one is hyperbolic in
the sense of Gromov since its automorphism group is cocompact and it does
not contain fiats. This will become more clear in (1.6) and Sect. 6 below.
1.6 Construction method. We dwell on a general method for constructing
2-orbihedra (X, F). With this method one can construct the examples in (1.5),
but it will also be important for another example below. The method is a slight
generalization of the corresponding one in [BB] and it relies on Theorem 2.3
in that paper.
1.6.1 Definition We say that a 2-orbihedron (X, F) is a symmetric pair /f F
contains a distinguished family o f elements (ae)eEex such that
(i) 0-e reflects the edge e and a2e = idx;
(ii) if f is a f a c e adjacent to e, then 0-e(f) : f',
(iii) 0-~,(e) = ~O'e~- 1 , Y E F.
We call 0-e the reflection along e. It is not clear whether such a family of
reflections is uniquely determined.
Choose a vertex v E X as origin of X and let f be a face with k edges
(a k-gon) adjacent to v. Let el and e: he the edges going out of v adjacent to
f and let ai = 0-ei- Note that
O, 0-11)70-10-2 I), G1 O"20-1V,...
are the vertices of f in a consecutive order. I f k = 2l we get
(1.6.2)
z(el,e2) := (0-10-2)l C F~,
z(el,e2)(el) = el,
z(el,e2)(e2) = e2,
where Fv C F is the isotropy group of F at v. I f k = 2l + 1, then
(1.6.3) z(el,e2) := (0-1a2)t0-1 E Fv,
z ( e b e 2 ) ( e l ) = e2,
z(el,e2)(e2) = el •
From Property (iii) we conclude
(1.6.4)
z(yel,ye2) = y'c(el,e2)y - l ,
y E Fv.
For k even we get z(el, e2)-1 = z(e2, el ) since the reflections are involutive.
Furthermore, for z = z(el,e2), we have z0-2z - l = o"2 since z(e2) = e 2 by
(1.6.2). Hence
1 = 0"10"2 "'" (0-1 z - 1 )('g0-2q: - l ) = 0-10-2"'" (0-1 "~-I )0"2
and therefore z(e2, el ) = z = z(el, e2).
Metrics on 2-polyhedra
105
For k odd we get "c(el,e2) -1 -~ z(el,e2) since the reflections are involutive.
Furthermore, for ~ = z(el,e2), we have ZalZ -~ = a2 since z(el) = e2 by
(1.6.3). Hence
1 ----ffl0"2''' (0"2~-I)(rO'l "f-I ) ----0"10"2 "'" (0"2"f-I )0"2
and therefore v(e2,el) = z = z(el,e2). In either case, k odd or k even, we
therefore have
(1.6.5)
z ( e l , e z ) - l = ~(el,e2) = z(e2,el).
We call z(el,e2) = z(e2,el) the twist along the face f . The face f is represented by an edge e in A = A~, and we consider z as a function on
EA, ~ : EA ~ Fv, the twist map at v.
1.6.6 Theorem L e t A be a 9raph with 9irth # > 3. Let d C Aut(A) be
a 9roup o f automorphism o f A and let z : EA --* A, k : EA ~ 1N be two
functions such that
(N) # ( k ( ~ ) - 2)/k(~) > 2
f o r all ~ E EA and such that f o r ~ = ~(~), ~ E Ea,
(S1) r~-I = T~,;
($2) ~7(~) = 7z~:7-~, 7 E A;
($3) ~ ( ~ ) = ~, z~(~l) = q if k(e) is even respectively z~,(¢) = q, z~(q) =
i f k(~) is odd, where ~, q are the vertices o f A adjacent to ~.
Then there is a unique simply connected 2-polyhedron X with a preferred
vertex v together with a 9roup F o f automorphisms o f X containin9 a f a m i l y
(ae)e~EX o f reflections such that
(i) A~, = A and the f a c e in X correspondin9 to ~ E EA has k(e) edoes;
(ii) the isotropy 9roup o f F at v is A;
(iii) z is the twist map at v.
The proof o f Theorem 1.6.6 is the same as the proof o f Theorem 2.5 in
[BB]. Therefore, we omit it.
1.6.7 Remarks. (a) F acts rioidly on X: if 7 E F fixes v and induces the
identity on Av then 7 = idx because 7 = 14 and A C Aut(A). Note also that F
is transitive on Vx since it is possible to move by reflections along sequences
o f adjacent vertices. In particular, F is cocompact and properly discontinuous
iff A is finite.
(b) Note that F is isomorphic to the extension o f A by the reflections
a¢ : = a~, ~ EVA representing the edge e going out o f v, modulo the relations
~ -- 1,
~(~)
--
yO'~])-1
for 7 E A
and
(~rCan) l = z~ respectively (rr~an)ttr~ -- z~
106
W. Ballmann, S. Buyalo
if k ( e ) = 2l respectively k(e) = 21+ 1. Note that z --- id is a possible choice if
all k(e) are even. Then we say that X is untwisted. In fact, in this case it is the
Cayley complex of the Coxeter group generated by the reflections a¢, ~ E Va.
Problem. Assume that A is finite. Does F have a subgroup F' of finite
index which acts freely on X? Since X is simply connected and of nonpositive
curvature, this is equivalent to the requirement that U is torsionless. The existence of a torsionless subgroup would follow if F would admit a faithful finite
dimensional representation over •.
(c) If A is the 1-skeleton of a given Platonic solid, A = Aut(A) and
k(e) =- k with k > 2tt/(# - 2) E {4,6}, where # is the girth o f A, then there
are precisely two possibilities for z, see [BB].
I f A is the 1-skeleton of the n-simplex or the n-cube, n > 4, and if
A = Aut(A), there is precisely one possibility for z; if it is the 1-skeleton of
the n-octahedron, there are precisely two.
In the case of the Platonic solids the two possibilities were mentioned in
Example 1.5.
(d) In a similar way one can also describe the pairs (X, F), where X is a
simply connected 2-polyhedron (satisfying (N)) and F is flag transitive on X.
2 The Gauss-Bonnet formula
Let X be a topological space and F be a countable group which acts
on X. Under certain circumstances one can define an Euler characteristic
X = x(X : F ) of X modulo (the action of) F, cf. [CG]. The real number ;~
is a F-equivariant homotopy invariant.
In our situation, (X, F) is a finite 2-orbihedron, that is, X is a 2-polyhedron
and F C Aut(X) is cocompact and properly discontinuous. I f S is a set of
representatives of the F-orbits of the calls of X, then
(2.1)
z ( X " F ) = E ( - 1 ) dims 1
151 '
where F~ is the stabilizer of s,
5 =
Ft,(s) = s}.
Note that IF~I < o~ since F is properly discontinuous.
The Euler number behaves multiplicatively under transition to subgroups
of finite index. In the case that X is contractible and F virtually torsionless,
x(X : F ) is an invariant of F.
2.2. Curvature measure. Let d be a piecewise smooth metric on the 2-orbihedron (X, F). We now define the curvature measure, a measure on the Borel
sets of X. For this purpose we choose a F-equivariant triangulation of X such
that d is induced by smooth Riemannian metrics on the simplices of X. Note,
however, that the curvature measure does not depend on the choice of the
Metrics on 2-polyhedra
107
triangulation. For the sake of simplicity, we only define the curvature measure of the open cells of X. The measure of an arbitrary Borel set is defined
accordingly. For a vertex v let
(2.2.1)
Z(v) = z(Av),
the Euler characteristic of the link A~, of v in X. For a face f adjacent to v,
denote by ~ ( v , f ) the interior angle of f at v. The complete angle at v (the
length or volume of Av) is
(2.2.2)
~d(v) = ~ ~(v, f ) ,
where the sum is over all faces f adjacent to v. The curvature measure of v
is then by definition
(2.2.3)
K a ( v ) = ( 2 - Z(v))~ - ~d(V).
For an edge e and a face f adjacent to e denote by x ( e , f ) the total
geodesic curvature of e with respect to f . That is, if kf denotes the geodesic
curvature of e with respect to f , then
(2.2.4)
x ( e , f ) = f kf .
e
The curvature measure of e is by definition
(2.2.5)
Kd(e) = ~ x ( e , f ) .
fie
For a face f denote by K the Gaussian curvature of f . Then the curvature
measure of f is by definition
(2.2.6)
Ka(f) = f K.
f
Assume now that (X, F ) is a finite 2-orbihedron and that d is a piecewise
smooth metric for (X, F). Then we define Kd(X : F), the total curvature of X
modulo F, by the formula
(2.2.7)
Ka(X" F) = Y] Ka(s)
s~s 1 5 t '
where S is the chosen set of representatives of the F-orbits of the cells of X.
2.3 Gauss-Bonnet Formula. I f d is a piecewise smooth metric for the finite
2-orbihedron (X,F), then
~ca(X : F) = 2rcX(X : F ) .
The proof is along the same lines as the usual proof for closed surfaces.
Indeed, except for a little bit of combinatories, we invoke only the standard
Gauss-Bonnet formula for the closed faces of X. The referee pointed out to
108
W. Ballmann, S. Buyalo
us that there is another combinatorial approach to the (higher dimensional)
Gauss-Bonnet formula in [Bri].
Proof of Formula 2.3. Separate the set S in the parts V,E and F corresponding
to vertices, edges and faces respectively.
2.4 SublemmaWe
have
K(e,f)
(i)
E E -
f~elf
(ii)
Irsl
-- Ot(v, f )
E E
151
fEFvif
(iii)
~ ~
-
E E
~c(e,f)
tr~l
e~Efie
= E E
z~ - ~(v, f )
<vii°
1
I~1
2
.~," el~ :vl ----<E It<l
This sublemma is trivial in the case F -- {id}. We finish the proof of (2.3)
first and prove (2.4) afterwards.
By the usual Gauss-Bonnet formula we obtain for each face f of X
2Tt = K ( f ) + ~ x(e, f ) + ~ (rt - ~(v, f ) ) .
elf
vtf
Hence, by Sublemma 2.4,
2n
x(f)
: ~ 151 = E
= E to(f)
:~,~iTrTl + eEE
E Efle
= ~
~(f)
fcR
rt - or(v,f )
x(e,f)
+ fEE r ~IS
E - - + E E fEE vif
-x (-e ,+f )
EErt-~(v'f)
trel
vCV fl~
Ir~l
~(e)
rt - o~(v,f)
"~'e~E~el "k- ~vsl~
E E
Irol
2/z
vEV
--
~v[
EV
+ E Irel"
Now note that for any vertex v of X,
Ird
Ird
Proof of Sublemma 2.4. The proof of the three formulas is along similar
lines. We only prove Formula (i) and leave the proof of (ii) and (iii) to the
reader. Set
A = { ( e , f , y ) i e E E, f i e a face, 7 E Ff}
B = { ( f , e , 7 ) l f E F, e t f an edge, 7 E Fe}.
Metrics on 2-polyhedra
109
For each edge e and face f of X fix ~e, Yf 6 F such that ])e(e) E E and
Y f ( f ) E F. Then
(p(e, f , 7) = (~f ( f ) , 7f7-1(e), 7f~- 1Y~,,fT-'(e ) )
$ ( f , e, 7) = (Te(e), ~e~-1 ( f ) , ~,)ey-l~/Te,~-l(f))
define bijections from A to B and B to A respectively. To see this notice for
example that
7yfy-l(e)(TfT-l(e)) = e for e 6 E
since e is the unique element of F(e) in E. Hence ~o(e,f,7) ig in B for
( e , f , 7) E A. Moreover,
~k(tp(e, f , Y)) = ~b(yf(f), yfT-I(e), yf'})--lT,/fy--l(e)) = (e, f', 7 ' ) ,
where
f / = 2;r~,-l(e)(YfY
-
-
1
Y~yT-~(e)) I ( ~ f ( f ) ) = 7Y71(Yf(f)) = 7 ( f ) = f
-
since 7 ¢ Ff and where, for e' = yfT-l(e),
71
-I
: 7et(Tf ~ yet )
-1
7yet(yly_17et)_l(Tf(f) ) = ,~yjl]yy(f) =
since 7 ( f ) = f . This shows t# • q~ = id, and similarly one shows q~ • ~ = id.
Now for (e, f , y) ¢ A we have
iv(e,f) : ~ : ( 7 - 1 ( e ) , y - l ( f ) ) = tc(y-l(e),f) = iv(TfT-l(e),~f(f))
since our metric is F-invariant. Hence
Iv(e, f )
f
ell
iv(e, f )
iv(e, f ) _ E ~ iv(e, f )
Irzl
Note that, in the last sum, edges in E which are not adjacent to any face,
contribute 0. Hence (i). []
2.5 Remark. The curvature measure of a vertex might be positive even though
the curvature of X is negative in the sense of Alexandrov.
3 Rigidity due to Gauss-Bonnet
Recall that a 2-orbihedron (X, F) is called closed if it is finite, if X is connected and if every edge o f X is adjacent to at least two faces. From now on
we assume that (X,F) is closed. This allows for a simple notation, and our
applications only concern closed 2-orbihedra anyway.
We say that an edge of X is essential if it is adjacent to at least three faces.
The other edges o f X are called inessential; these are adjacent to exactly two
faces each since X is closed. We say that a vertex v of X is inessential if Av is
110
W. Ballmann, S. Buyalo
homeomorphic to a bipartite graph with two vertices and n = n(v) ~ 2 edges
connecting them. The other vertices of X are called essential. An inessential
vertex v of X is called an interior vertex if n(v) = 2.
A connected component of the union over all faces, inessential edges and
interior vertices is called a maximal surface of X. A connected component of
the union over all essential edges and inessential vertices v with n(v) > 3 is
called a maximal essential edge. A maximal essential edge might be a loop.
Since (X,F) is closed, the maximal surfaces of X are bounded by maximal
essential edges and essential vertices.
Recall the separation of the chosen set S of representatives of F-orbits of
cells in X into the sets V, E and F corresponding to vertices, edges and faces.
Denote by Ve and V,- the set of essential respectively inessential vertices in V.
For a given piecewise smooth F-invariant metric d on X set
(3.1)
va(X: F ) : = ~
~ca(v)
- -
+
g-, xd(e)
+
~:a(f)
Then by definition
(3.2)
xa(X : F) -~ ~
Ka(v__~)+ va(X" C).
3.3 Theorem Assume that d & a piecewise smooth metric for (X, F) with
nonpositive curvature. Then va(X : F) < 0 and equality holds iff
(i) all maximal surfaces of X are fiat;
(ii) all maximal essential edges o f X are geodesics.
Proof Since d has an upper bound on the curvature (namely 0), we can
apply Theorem 7.1 from the Appendix. Hence, whenever faces f~ and f2 are
adjacent to an edge e and x C Int(e) is smooth with respect to fl and f2,
then kfi(x) + kf2(x) < O, where k~ denotes the geodesic curvature of e with
respect to fi, i = 1,2. Moreover, i f x is one of the finitely many points in the
interior of e where e is not smooth with respect to .fl or f2, then
cX(X,f l ) q- ~(x, f 2 ) >= 2re,
where ~(x, j~) denotes the interior angle o f e at x with respect to J~, i = 1,2. In
particular, the contribution o f the edges to Vd(X : Y) is nonpositive. Again since
d has an upper bound on the curvature, simple loops in the links A~, v E Vx,
have to have length at least 2n with respect to the length metrics induced
by d, see Theorem 7.1. Hence the contribution of the vertices in I1,- is nonpositive. Since the Gauss curvature of any face has to be nonpositive, the
contribution of the faces to va(X : F) is nonpositive as well, hence Vd(X :
F ) < 0. Now vd(X : F) = 0 implies that the contribution of the vertices,
edges and faces to vd(X : F) are 0 each. Hence the Gauss curvature of the
Metrics on 2-polyhedra
111
faces has to vanish. For e, f l ,
(*)
f2
kA(x)+kfz(X ) = 0
and x as above we must have
resp.
e(x, f l ) + e ( x , f z ) = 2n.
Furthermore, simple loops in the links A~, v E Vi, must have length exactly
2n. This implies (i). If there are at least three faces f l , f 2 , f 3 .... adjacent to
e and if x is smooth with respect to all o f them, and kf~ (x) = fl > 0 say, then
k f 2 ( x ) = k f 3 ( x ) = --fl
and hence
kfz(x ) + kf3(x ) < O,
a contradiction to (.). A similar argument also applies to e ( x , f ) if x is not
smooth with respect to some face, and shows e ( x , f ) = n. We conclude that e
is a geodesic. Similar arguments show that any edge in the link Av has length
exactly n if v is inessential and n(v) > 3. For this one uses that all simple
loops have length exactly 2n. Hence essential edges connect up to geodesics
at inessential vertices v with n(v) > 3. This shows (ii). []
In our applications o f Theorem 3.3 we need estimates on ~:d(V), v EVe, so
that the Gauss-Bonnet formula ensures rigidity phenomena. Recall that Kd(V)
was defined as
~ca(v) = ( 2 - x ( & ) ) ~
- ~d(v)
where Av is the link o f v in X, z(Av) is the Euler characteristic o f A~. and
Ctd(V) is the length o f A~ with respect to the length metric ctd on Av induced
by d. Now, if A is any finite graph, and ~ is any length metric on A, denote
by c~(A) the length o f A and by sys(e), the systol o f ct, the length o f a shortest
essential loop in A. Recall sys(ad) > 2n for ad and A~ as above. Set
(3.4)
a(A) : = inf e(A)
sys(e)
the size of A and
(3.5)
x(A) : = (2 - z(A) - 2a(A))n,
the maximal total curvature o f A. Note that
(3.6)
tc~(A) : = (2 - z(A))n - ct(A) < re(A)
for any length metric a on A with sys(~) > 2n. A length metric ~ on A
with sys(~) > 2n and a(A) = 2ha(A) is called minimal; for these we have
~:~(A) = n(A). We say that a length metric for (X,F) is tight if
(3.7)
all maximal surfaces o f X are flat
)
all maximal essential edges o f X are geodesics
for any essential vertex v of X, ed is minimal
Every tight metric is nonpositively curved. We have the following topological
criterion for tightness.
112
W. Ballmann, S. Buyalo
3.8 Theorem Suppose that (X, F) is closed and admits a piecewise smooth
metric of nonpositive curvature. Then
~(Av)
v~v,
Ir~l
> 2~z(x
=
: r)
and equality holds if and only if one, and hence any, piecewise smooth metric
for (X,F) with nonpositive curvature is tight.
This follows immediately from Gauss-Bonnet, Theorem 3.3 and the definition of x(A).
3.9 Corollary Suppose F r C F is a subyroup of finite index. I f (X, F) admits
a tiyht metric d, then any piecewise smooth nonpositively curved metric jor
(X,F') is tioht.
This follows immediately from Theorem 3.8 since d is tight for (X,U). To
apply Theorem 3.8 we need to know the maximal total curvature of graphs.
We prove a first result in this direction; more detailed information can be
found in Sect. 4. Recall that the 9irth of a graph A is the maximal natural number # such that any essential loop of edges in A has at least #
edges.
3.10 Proposition Let A be a finite #raph with 3 vertices, e edges and 9irth
lz > 2. Assume that there is a number 2 > 1 such that any edge of A is
contained in exactly 2 essential loops with p edges. Then
cr(A)=-~a n d #
x(A)=(2-6+~
~)Tr.
Proof Let C be the set of essential loops in A with # edges. Then IcI -- 2~/~.
If ~ is any length metric on A with sys~(A) > 2~, then
~ ( A ) = ~ l~(c) >= 2~ 2~
cEC
#
where l~(c) denotes the length of c with respect to ~ and hence ~(A) => 2roe,/#.
On the other hand, equality is obtained for the length metric attaching length
27t/~ to every edge of A. []
3.11 Examples. If the automorphism group of A is transitive on the edges of
A, then the number 2 > 1 as required in Proposition 3.10 exists.
(a) If A is the 1-skeleton of one of the Platonic solids, then x(A) = O.
(b) If A is the 1-skeleton of the
i) n-simplex, then to(A) = (n - 2)(n - 3)7z/6;
ii) n-cube, then x(A) = ((n - 4)2 n-2 + 2)re;
iii) n-octahedron, then x ( A ) = 2 ( n - 1 ) ( n - 3)rc/3.
In particular, in each case ~(A) > 0 for n > 3.
Metrics on 2-polyhedra
113
(c) If A is the 1-skeleton of the truncated (at the midpoints of edges) cube,
then x(A) = -27t. If A is the 1-skeleton of the truncated icosahedron, then
It(A) = --8m
Say that a polyhedron X is simple (without boundary) if the link of any
essential vertex of X is homeomorphic to the 1-skeleton of the tetrahedron and
the link of any inessential vertex of X is homeomorphic to the bipartite graph
with two vertices and two or three edges connecting them.
3.12 Corollary (i) If X is simple and if (X, F) has a piecewise smooth metric
of nonpositive curvature, then z(X : F) < O.
(ii) More 9enerally, if (X,F) is closed and the link at each essential
vertex of X is homeomorphic to the 1-skeleton of one of the Platonic solids,
and if (X, F) has a piecewise smooth metric of nonpositive curvature, then
z ( x : F) __< o.
In either case, equality implies that any piecewise smooth metric of nonpositive curvature is ti#ht.
3.13 Examples. (a) If A is a finite graph with 6 vertices and e edges, and if
X is a polyhedron such that the link at each vertex of X is isomorphic to A
and such that each face of X is a p-gon, that is, it has p > 3 edges, then for
F as usual
2z(X : F) =
(
~)v~v
2 - 3 + --
1
Irv[
'
compare the proof of Sublemma 2.4. If the number 2 as in Proposition
3.10 exists, then we have equality in Theorem 3.8 iff p = 2 # / ( / ~ - 2),
that is, if
/ ~ = 3 and p = 6
or
#=4
and
p=4
p=6
and
p=3.
or
In this case the metric d on X such that any face of X is a regular Euclidean
p-gon with side length 1 is tight. If p > 2#/(# - 2), then X admits a metric
with negative upper bound on the curvature. For the construction of polyhedra
X as in this example see [BB, Be, Ha] and also (1.6).
(b) For every k > 3 and n => 3 there is a finite connected polyhedron
Y such that all faces of Y are 2k-gons and the link of each vertex in Y is
isomorphic to the 1-skeleton of the n-simplex, cf. Examples 1.3.b and 1.6. If
we let d be the length metric on Y such that any face is isometric to the
regular Euclidean or hyperbolic 2k-gon with interior angles 2n/3, then d is
nonpositively curved for k = 3 and negatively curved for k > 4. The Euler
characteristic is
z(Y) = (1
n+l
n(n + 1 ) )
- --~ +
4~
v,
114
W. Ballmann, S. Buyalo
where v is the number of vertices of Y. Hence z(Y) > 0 for n > 2 ( k - 1)
despite the fact that the curvature is negative. From Corollary 3.12 we conclude
that no simple 2-polyhedron Z homotopy equivalent to Y admits a piecewise
smooth metric of nonpositive curvature. This fact is interesting because strictly
negative curvatm'e is, in a sense, an open condition and, on the other hand, Y
is the Hausdorff limit of simple polyhedra.
Question: Let Y be an aspherical finite 2-polyhedron with z(Y) < 0. Does
there exist a simple 2-polyhedron Z homotopy equivalent to Y such that Z
admits a metric of nonpositive curvature?
4 Minimal metrics on graphs
In order to discuss the space of tight metrics on a 2-polyhedron, we need
information on minimal metrics on graphs.
4.1 Labelinos and metrics on 9raphs. Let A be a finite graph and E
= EA
the set of (nonoriented) edges of A. If d is a length metric on A, then, up to
natural equivalence, d is given by
(4.1.1)
I:E--.IR,
e~-~l(e),
where l(e) denotes the length of e with respect to d. This function
will be called the labelin9 (of the edges of A). All we need to know
about d is contained in this function. The space of labelings on A is IRe+.
As a normalization, we require that sys(l) > 27t. This condition amounts
to a finite set of linear inequalities (given by the simple loops of edges) on
the space IRe+ o f labelings on A, hence
LA -----{l E IRE+lsys(l) > 2n}
is a convex subset of IRe+. The length functional
(4.1.3)
l(a) = ~ l(e)
eGE
is linear in l E IRe+. In accordance with the terminology in Sect. 3 we say that
l is a minimal labelin9 if
I(A) = inf{f(A)l f
E LA} = 2~o'(A) .
As the intersection o f a convex set with a hyperplane, the set o f minimal
labelings
(4.1.4)
MA := {1 E LaII(A) = 2rw(A)}
is a convex subset of La and IRe+. Note that MA may be empty, even if A is
not simply connected, compare Fig. 2.
Metrics on 2-polyhedra
115
71"
Fig. 2
7r--~
71---4
Fig. 3
Let e be an essential loop o f edges in A. Say that
for any l E MA. A principal loop has to be simple,
be principal, even if it has length 2n for some l E
A set C o f principal loops is said to be ample
(4.1.5)
c is principal if l(c) = 2n
but a simple loop need not
MA, compare Fig. 3.
if
a labeling l E LA is minimal if l(c) = 2n for any c E C;.
Notice that an ample set o f loops cannot be empty if MA 4=O. Indeed we have
(4.1.6)
if MA+-O and C is ample, then C covers A .
Proof Let l E MA, and assume the edge e is not contained in any loop o f
C. Then we can increase the length o f e, keeping the metric in MA. This is a
contradiction.
[]
4.1.7 Proposition If MA 4=O, then
(i) the set CA of all principal loops is ample and, in particular, not empty;
(ii) d i m g a = IEI - rank (CA),
where rank (CA) is the maximal number of linearly independent hyperplanes
{l(e) = 2role E CA} in IRE+.
Proof Let l E MA and set
Ct = {clc is a loop o f edges in A and
l(c) = 2rt}.
116
W. Ballmann, S. Buyalo
Then every c in Ct is simple. Let
Ql={fElRE[f(c)=2n
for a l l c E C t } .
Then l E Qt. Furthermore, since l(c) > 2n for any loop of edges c in A
with c ~ Ct, an open neighborhood U of l in Qt belongs to LA. Now
the linear functional f ~ f ( A ) achieves a minimum in the open set U of
the affine space Qt, namely at l since l E MA. Hence f ( A ) = l(A) for all
f E Q1.
Suppose now that 1 is in the interior of the convex set MA. Then any curve
c E Ct is principal. Otherwise there would be an f E MA with f ( c ) > 2n.
But then also an f E Ma with f ( c ) < 2n since l E Int(MA). This is a
contradiction because M A C La. Vice versa, if c E CA, then c E Ct by the
definition of CA. This shows CA = C1. Since the length functional is constant
on QI, we conclude MA = Ql N LA, hence (i). Since a neighborhood of l in
Qt belongs to La (and hence MA), the interior of MA in Qt is not empty.
Therefore dimMA -= dim Qt, hence (ii). []
For the sake of reference we state the following slight generalization of
Proposition 3.10.
4.1.8 Proposition Let C be a set of essential loops o f edges in A such that
(i) there is a 2 > 1 such that every edge in A is contained in exactly ,~
loops from C;
(ii) there is an lo E LA such that lo(c) = 2re for all c E C.
Then lo is minimal with lo(A) = 2rtlCt/2, every c E C is principal and
C is ample. []
4.1.9 Remarks. (a) Under the assumptions of Proposition 3.10 we can choose
C to be the set of all essential loops with # edges and l0 - 2n//~. Observe that
C is the set of all principal loops because lo(c) > 2n for any other essential
loop.
(b) Suppose a labeling lo E La has two essential loops cl and c2 of length
2~ such that
(i) ci and c2 have an edge e0 in common;
(ii) cl has an edge el and c2 an edge e2 which are not contained in any
other essential loop of length 2~.
Then l0 is not minimal. In fact, there is a labeling l with I(A) = lo(A)
and sys(l) > 2re. This shows that the conditions in the proposition are close
to being necessary for the set C to be ample.
Let A be a finite graph such that each vertex is adjacent to at least two
edges, and let do be the metric on A which assigns to each edge of A the
length 1. Then A is a Tits building (spherical of rank 2) iff there is a number
m >__ 2 such that diam(A) = m and girth(A) = 2m, see [Bro], Exercise 2 on
p. 87.
Metrics on 2-polyhedra
117
4.1.10 Proposition I f A is a Tits buiMing of girth 2m, then the set of simple loops of 2m edges is ample. In fact, every edge of A is contained in
2 = 1 - z ( A ) such loops.
Proof Let e be a (closed) edge of A, and adjoin to e all points which lie on
paths o f k closed edges, k < m - 1, starting from one of the end points o f
e. The union T o f e with all these points is a tree since girth(A) = 2m. On
the other hand, since the diameter o f A is m, we have A = T U S, where S is
the set o f points on edges adjacent to T. Each such edge f has its endpoints
in T and gives rise to a simple loop o f 2m edges through e. Hence there is a
one to one correspondence between these loops and edges f in A - T. Since
z ( T ) = 1, the number of such loops is 1 - ~(A) as asserted. D
4.2 Minimal labelings and metrically rigid graphs. A finite graph A is said to
be metrically rigid if there exists only one minimal labeling on A. According
to Proposition 4.1.7, this is equivalent to MA ~ 0 and rank (CA) = lEA t.
In what follows we discuss minimal labelings and rigidity o f some standard
graphs. We will use the results o f Propositions 4.1.7 and 4.1.8 and Remarks
4.1.9.
4.2.1 Bipartite graph with two vertices and n > 1 edges. In this example, A
consists o f two vertices and n edges connecting them. This graph occurs as the
link of inessential vertices. If n = 1, then A does not have minimal labelings.
For n > 2, the labeling l0 = 7r/2 is minimal. For n = 2 we have dim M A ~- 1,
for n > 3 we have that A is metrically rigid.
4.2.2 1-skeleton of 3-simplex. Let A be the 1-skeleton o f the 3-simplex. Then
the labeling l0 on A defined by lo(e) -- 2rt/3 for all e E EA is minimal and
the set o f principal loops consists of the 4 simple loops of 3 edges.
For any minimal labeling l o f A we have l(e) = l(~) for any pair o f
opposite edges e, ~ of A. The space Ma of minimal labelings is an equilateral
triangle minus the vertices; the (open) edges correspond to labelings where one
o f the 3 simple loops o f 4 edges has length 2rt, cf. Fig. 3 above.
4.2.3 1-skeleton of n-simplex. Let An be the 1-skeleton o f the n-simplex, n >
4. Then the labeling 10 on AN defined by lo(e) = 2rr/3 for all e E EA. is
minimal. Moreover, An is metrically rigid.
Indeed, every pair e, Y o f opposite edges in An spans a 3-simplex. Hence
l(e) = l(~) for any minimal labeling l, see (4.2.2). If e,e J are not opposite,
there is an edge ~ opposite to both e and e' since n _> 4. Hence l(e)= l(e')
and therefore l -~ 2~/3, that is, l = 10.
4.2.4 1-skeleton of n-cube. Let A, be 1-skeleton o f the n-cube, n > 2. Then
the labeling l0 on An defined by lo(e) = 7t/2 for all e E EAn is minimal. For
all n > 2, A, is not metrically rigid.
Indeed, think o f the vertices o f the n-cube as the vectors v = (vl . . . . . v,)
in IRn with vi E { - 1 , 1}, 1 < i _< n. If v,w are such vertices, then there is an
118
W. Ballmarm, S. Buyalo
edge vw between v and w iff v - w = ±2ei, where ei is some vector from the
standard basis o f IRn. Suppose now that l is a minimal labeling of the n-cube
such that, for some A E O(n) leaving the n-cube invariant, we have for all
edges vw
(,)
l(A(vw))
re_
2
(l(vw)_ 2)
Then we can define a labeling It of the edges o f the ( n ÷ 1 )-cube as follows:
ll(vw) ~-
l(vw)
if
l(A(vw))
and
if
zt/2
and
if
v w = + 2 e i , 1 <- i <_ n , ]
/)n+l ~ Wn+l ~---1;
vw = ±2el, 1 <-- i <_ n
v,+l = Wn+l = +1;
vw = ± 2en+l.
J
Then l' is a minimal labeling since all the essential loops of four edges
have length 21r. Moreover, if we extend A to A' E O(n + 1) by demanding At(en+l ) = (e~+l) then It,A ' satisfy (*). Hence starting from the minimal
labeling
-~, ~-& ~+~, ~+~ , 0<~,,~<~,
for the 2-cube and A the reflection about 0 in the plane we get recursively
non-trivial minimal labelings for all n-cubes, n >= 3.
Question. What is dim MA.
4.2.5 1-skeleton ofn-octahedron. The n-octahedron is, by definition, the dual of
the n-cube. Now let An be the 1-skeleton of the n-octahedron. Then the labeling
10 on An defined by lo(e) = 2rc/3 for all e E Ea, is minimal. Moreover, A, is
rigid for n > 5.
Indeed, every edge of An lies in the 1-skeleton of an (n - 1)-simplex
Atn_l C An; hence An is rigid for n > 5 by (4.2.3).
It is easy to see that A, is not rigid for n < 3; a somewhat tedious
computations shows that A4 is not rigid.
4.2.6 1-skeleton of the dodecahedron. If A is the 1-skeleton o f the dodecahedron, then the labeling l0 on A defined by lo(e) = 2rc/5 for all e E EA is
minimal. We have dim MA = 18. To show the latter claim, choose a vertex
v0 in A and the following three simple paths e1,c2,c3 of six edges: each ci
starts at v0, turns left at the first vertex, left at the second, then right, left, right
respectively. It is easy to see that for given numbers l l , . . . , 118 close to 2rc/5
there is a unique minimal labeling o f A attaching these numbers as labels to
the 18 edges of c~, c2 and c3.
4.2. 7 1-skeleton of the icosahedron. I f A is the 1-skeleton of the icosahedron,
then the labeling 10 on A defined by lo(e) = 2zc/3 for all e E EA is minimal.
We have dim Ma = 10.
Metrics on 2-polyhedra
119
Fig. 4
The proof is similar to the one in the case of the dodecahedron and left as
an exercise.
Let A be a finite graph which is a Tits building (see the
remarks above Proposition 4.1.10 for the definition). We say that A is thick if
every vertex of A is adjacent to at least three edges. For example, if A is the
flag complex of the projective plane over a finite field K, then A is a thick
Tits building of girth 6. In the case K = E/2, this graph is called H e a w o o d
graph, see Fig. 4. We have:
If A is a thick Tits building of girth 2m, then A is metrically rigid with
minimal labeling Io - n/m.
4.2.8. Tits buildings.
Proof. Let c = el *" - "*e2m be a simple loop of 2m edges. Since A is thick, there
is an edge f l starting from the endpoint of em and not contained in c. Since A
has diameter m with respect to the metric do which assigns length 1 to every
edge of A, any point on f l has distance at most m to the initial point of el.
Hence there must be a segment f 2 * ' " "*fro of m - 1 edges connecting the end of
f l to the initial point of el. Since the girth of A is 2m, f l *'" "*fro intersects c
only in the endpoint of em and the initial point of el. Hence A' = c t 3 ( f l , . . . .
fro) is homeomorphic the bipartite graph with two vertices and three edges.
Now a minimal labeling l of A assigns length 2n to any simple loop of 2m
edges, see Proposition 4.1.10, hence its restriction to A ~ is minimal. By (4.2.1),
el * ... * em has length n. We conclude that any simple path of m edges in
A has length exactly n, in particular, e2 * . . . * era+l, and e2 * ..- *em * f l .
Hence em+l and f l have the same length (that is, the same label). Therefore,
the labeling l is constant on the edges emanating from a given vertex of A,
hence l is constant. []
Let A = Am.n be the join of two sets consisting of m and n points
respectively, m , n >= 2. Then the labeling l0 on A defined by to(e) = n/2 for
all e E Ea is minimal. If m,n -> 3, then Am, n is a thick Tits building of girth
4, and hence Am, n is rigid.
4.2.9 Join.
120
W. Ballmann, S. Buyalo
4.2.10 Join minus one edge. Let A be the graph obtained from the join A3, 3 by
removing one edge. Then A is obtained from the 1-skeleton of the 3-simplex
by subdividing an opposite pair o f edges. Hence dim MA = 4.
4.3 Collision o f graphs. We describe a procedure which provides a new graph
with a minimal metric, starting from two graphs with minimal metrics. It is
the most general construction o f minimal metrics on graphs which we know
of. Moreover, it reflects exactly the result o f a collision o f the two ends o f an
edge during a deformation through tight metrics on a 2-polyhedron collapsing
this (but no adjacent) edge, see Theorem 5.11.
N o w let A ~, A '1 be the graphs, x' E A ~ and x" E A" vertices with the same
number k o f adjacent edges e] .... ,e~ E EA, and e~'. . . . . e~t E EA,,. As usual
we assume that none o f these edges is a loop. Denote by x[ E A' respectively
x~' E A " the end of e~ opposite to x I respectively o f e~~ opposite to x". Define
the graph A = A ( A 1, A 'I, (e~), (e~')), the collision o f A I, A '1 at xl,x '1, as follows:
(4.3.1)
VA = ( V A , \ { X ' ) ) U
EA = (EAt \ {e~ . . . . .
U ( E A,, \
U {el,...,ek},
I. . . .
where the edge e; connects x;' E A' to x i" in A". This is well defined since
none o f the edges e i, e i" is a loop.
Suppose now that l ~ E LA,, l" E Lit,, satisfy
l'(e~)+ l I I (e iI t ) > ~,
(4.3.2)
i = 1. . . . . k .
Define the collision labeling 1 on A by
l'(e)
(4.3.3)
l(e) =
if e E E A N E A ,
ll'(e)
if e E EAf-) EA,,
l'(e;) + l"(e~') -
if e - - e i ,
i-----1 . . . . . k .
4.3.4 L e m m a I f l' E LA,, 1" E LA,, satisfy (4.3.2), then l E LA, then is,
sys(l) > 2n.
P r o o f It is sufficient to check that the length o f a sitfiple loop c containing at
least one o f the edges el . . . . . ek is at least 2zt. It is then clear that c contains
an even number 2m o f those, and without loss o f generality we assume it
contains el . . . . . e2m. N o w let c i be the part of c in A ~ between ei and ei+l,
i = 2,4 .... , 2 m ( i m o d m ) and let cy be the part o f c in A" between ej and
e j + b j = 1 , 3 , . . . 2 m - 1. Then
m
l(c)=
i=I
' + l'(c¢2i) + t1¢e
I ~
( l ' (e2i)
i 2i+11)
m
+ X-~
Z . ~t ltt l~e~t
2j-ly.~ +, l t/ ( e tl2 j - l ) + l " ( d 2 ~ ) ) - 2 m r ~
j=l
_-> 2 m 2 x - 2 m x
= 2mx > 2 ~ .
[]
Metrics on 2-polyhedra
121
1
2'
~,7
3"
4n ~
4
Fig. 5
4.3.5 Example. If A is the collision of A',A" and l~,l" satisfy (4.3.2),
then l need not be minimal even if l p and I" are, see Fig. 5. Below in
Proposition 4.3.6 we will obtain a sufficient condition for l to be
minimal.
For k > 3 let Zk be the set of all finite graphs A such that
(i) all vertices of A have valence k;
(ii) there is a set Ca of essential loops in A such that for any two adjacent
edges e , # in A there is a unique c ( e , # ) E CA containing e,#;
(iii) there is a labeling l E LA such that l(c) --= 2n for all c E CA.
It follows from (i)-(iii) that all c E CA are simple, no edge of A is a
loop (recall k > 3) and every edge of A is contained in exactly k ( k - 1)/2
loops from CA. Thus Proposition 4.1.8 applies and shows that CA consists
of principal loops and is ample. Moreover, l as in (iii) is minimal and each
minimal labeling satisfies (iii).
Denote by S~ the subset of A in ~k such that
(iv) CA can be chosen as the set of all principal loops of A.
This condition allows to compute the dimension of the space MA of minimal
labelings, see Proposition 4.1.7,
dim M A
=
IEAI -- rank(CA ).
Observe that the 1-skeleton of the k-simplex and the k-cube are in Z~,
the l-skeleton of the k-octahedron is in 2~(k_l) and the 1-skeleton of the
dodecahedron is in Z~. The 1-skeleton of the icosahedron is not in Zs.
4.3.6 Proposition The sets ~ , ~ are closed under the collision operation
whenever condition (4.3.2) is satisfied for some minimal metrics. Furthermore,
collision labelings are minimal.
Proof Let X , A " E 2~k collide at x p EVA,, x" EVA,,, and define l as in
(4.3.3). Let CA consist of those c E CA, respectively CA,, which do not meet
x ~ respectively x" and of the loops cij, 1 <= i < j < k, defined as follows:
the edges e~, #.. respectively e~t, ey define unique elements c(e i,~ e))
' = cq' E Ca,,
. . , e))
. . _ - cij
h E Ca,, and we take cij the collision of cij
t a n d cij.
" Then we
c~ei
122
W. Ballmann, S. Buyalo
have for the collision labeling 1
l(cij) = l , (cij)
, + l ,, (ciy)
,, - 2re = 2rt,
hence A satisfies (i),(ii),(iii).
Now assume A ' , A " satisfy (iv) and let c be a principal loop in A.
If c does not contain any of the edges el . . . . . ek, then c is contained in
A', say, and has length 21r for all minimal labelings on A t such that
condition (4.3.2) holds. Since this is an open condition, c is a principal
loop o f A', hence in C~ and therefore in CA by the definition
of CA.
I f c contains an edge ei, 1 < i < k, then it contains an even number
2m of these edges and we have l(e) > 2mrc as in the proof of Lemma
4.3.4. Therefore rn = 1 and c is obtained as the collision of loops c t, c" which
have length 27r for all l', l" satisfying condition (4.3.2). Hence c t, c" are principal since this condition is open. Hence c' E CA,, c" E CA,, and therefore
C C CA.
[]
4.3.7 Proposition For A t, A" C Z~ and A a collision of A t, A", we have
0 =< dimMa - (dimMA, + dimMA,,) < k(k - 3)
2
Proof For CA, Ca, and GAit
as
above, the matrix representing C A has the form
0/
R
B"
0
A"/
,
where
Bp
and
=
A"
corresponds to the equation defined by CA, respectively CA,, and where
(B'RB") represents the equations defined by the loops cij E CA as above,
that is, R is the (kz) × k-matrix
R=(RP),
1 <i
< j _ _ < k and 1 < p__< k,
with
R P = [" 1~0 i f p = i o r
p=j;
otherwise.
Now rank (R) > 3 since k > 3 and we can use R for elementary matrix
manipulations of C. We obtain a matrix
~=
o o)
I
0
~b~Ob""
0 A"
Metrics on 2-polyhedra
123
where I is the (k × k)-unit matrix and
dim ker
\b'
i)
= dim MA,,
dim ker
and dim ker C = dim MA. Hence the assertion.
(i /
if/
= dim Ma,,
[]
5 The space of metrics of nonpositive curvature
Let (X,F) be a closed 2-orbihedron. We denote by J-(X,F) the space of
equivalence classes of F-invariant piecewise smooth metrices of nonpositive
curvature on X, where two such metrics are defined to be equivalent if they
differ by a F-equivariant homeomorphism of X which is F-equivariantly isotopic to the identity. Any such homeomorphism leaves each essential vertex
and essential edge invariant. We endow ~--(X, F) with the quotient topology of
the uniform topology on the space of all metrics of X.
Our first interest is the dimension of J-(X, F). To that end we set
(5.1)
v(X : F) := inf{va(X : r ) l d E ~-(X : r ) } ,
where va(X:F) is defined in (3.1). Then v ( X : F ) < 0 if and only if (X,F)
has a piecewise smooth metric of nonpositive curvature. We also have
(5.2)
2r~z(x : r ) -
~ ~(A~) _-<v(X : r ) ,
hence v(X: F) > -cx~. The inequality in (5.2) can be strict, see Example
5.5(b).
By definition, v(X : F ) = 0 if and only if (X,F) admits a piecewise
smooth metric of nonpositive curvature and if, with respect to any such metric,
all the maximal surfaces of X are fiat and all the maximal essential edges are
geodesic. Then J ( X , F) is finite dimensional and, in fact, can be described as
a subanalytic subset of some ~U. Below we will study this in some particular
cases. On the other hand, we have the following result.
5.3 Theorem If v(X : F) < O, then J-(X, F) is infinite dimensional
Proof Let d be a piecewise smooth metric of nonpositive curvature on (X, F )
with va(X : F) < 0. Choose a F-invariant triangulation of X such that d is
induced by smooth Riemannian metrics on the closed faces of X.
Suppose first that there is a point x in the interior of a face of X such that
the Gauss curvature is negative at x. Then we can F-equivariantly deform d
in a neighborhood of the orbit F(x) in an arbitrary way, only subject to the
condition that we keep the Gauss curvature negative. Hence J ( X , F) is infinite
dimensional in this case.
We can assume now that with respect to any pieeewise smooth metric of
nonpositive curvature (and corresponding F-invariant triangulation of X), all
124
W. BaUmann, S. Buyalo
the faces of X are flat. Suppose, however, that for the given metric d and the
given triangulation of X, an edge e contributes to vd(X" F). That is, there is
a point x in the interior of e such that
kf(x) < 0. Recall that
~fle
(*)
kfl(x ) + kf:(x) < 0 for all f l :~f2
adjacent to
e.
We consider several cases.
Case 1. There is a face f 0 adjacent to e such that kfo(X ) < 0 and such that
kfo(x ) + k/(x) < 0 for all other faces f adjacent to e.
We want to show that Case 1 cannot occur. Note that the condition on x is
open. By subdividing the given triangulation of X if necessary, we can assume
klo < 0 and kfo + kf < 0 along all of e and that the two other edges of f0
are geodesic segments of length a respectively b. We denote by c the length
of e. Since f0 is flat, we can think o f f0 as contained in the Euclidean plane.
Now a comparison argument shows that the triangle fd in the Euclidean plane
bounded by two line segments of length a respectively b and a third edge of
length c with curvature k (not necessarily constant) such that kfo <_ k < 0
and kyo(X ) < k(x), has interior angles strictly larger than the interior angles
of f0. We choose k such that k + kf < 0 for all other faces f adjacent to e.
If e < 0 is sufficiently close to 0, the comparison triangle f~ of f ~ in the
hyperbolic plane of curvature ~ still has interior angles bigger than the ones of
f0. Hence we may replace f0 by this triangle, keeping nonpositive curvature.
This is a contradiction.
Case 2. If fo is such that kfo(X ) is minimal, then k/o(x ) < 0 but kfo(x ) +
kfl(x ) = 0 for some face f l adjacent to e.
Since kf(x)+k/l(X ) < 0 for any face f adjacent to e, f + f l ,
from (1), since kfo(X ) is minimal,
(**)
we conclude
kf(x) + kft(x ) -= 0 for all such f .
Since Case 1 is excluded, this equality also holds for all y on the edge e
close to x. Now think of f l and the other faces f adjacent to e as embedded
in the plane. By (**), f l and any other f precisely fit together along e in a
neighborhood of x. Hence we can (jointly) deform e in a neighborhood of x
in an arbitrary way and, therefore, J-(X, F) is infinite dimensional.
We can assume now that, with respect to any piecewise smooth metric
of nonpositive curvature (and corresponding F-equivariant triangulation of X),
all the faces of X are fiat and the curvature measure of the edges is zero.
Then, for the given metric d and the given triangulation of X, an inessential
vertex x of X has negative curvature measure. It is convenient to disregard the
given triangulation and think of a neighborhood of x as the union of finitely
many faces along a common edge e containing x in its interior (recall that x
is inessential).
Metrics on 2-polyhedra
125
Fig.
6
Case 3. There is a face f0 adjacent to x such that the interior angle ~(x, f0)
satisfies o~(x,fo) > n and ~(x, f o ) + ~ ( x , f ) > 2n for any other face f adjacent
to x.
From f0 we cut out two small geodesic triangles as in Fig. 6, where
-
2
< fl < min
Then the Euclidean triangle A t with side lengths b' = b, c ~ = c and a' slightly
bigger than a has interior angle/~' slightly smaller than fl and the other interior
angles strictly bigger than the ones of the given triangles. As in Case 1 we
conclude that we may replace each of the given two triangles with the comparison triangle At of A' in the hyperbolic plane of curvature ~ < 0, keeping
the curvature nonpositive. This construction can be done F-equivariantly, and
we obtain a contradiction to our assumption. Hence Case 3 does not occur.
Case 4. There is precisely one face fl adjacent to x such that ~(x, f l ) +
~(x, f ) = 2re for any other face f adjacent to x.
In this case, the faces f fit precisely to fi and we can move their common
boundary jointly, bringing us back to Case 2. This is a contradiction to our
assumptions, hence the Theorem follows.
[]
5.4 Remark. The arguments also show that the deformation space of a piecewise smooth metric d of nonpositive curvature with vd(X : F ) < 0 is infinite
dimensional•
The above proof needs only minor modifications. Namely, in Case 1 the
choice of the curvature function k is arbitrary, except for F-equivariance and
kfo <~ k < O, k:~kfo. In Case 3, the Euclidean comparison triangle A n with a
geodesic side of length b, a side of length c with negative geodesic curvature
k (not necessarily constant) close to 0, angle between these sides as in one of
the given two triangles (a, b, c), and third side a ~ geodesic, has interior angle
fl~ slightly smaller than fl and angle between a ~ and b bigger than between
a and b. Hence any such function k defines a deformation of d. The Cases 2
and 4 are clear.
126
W. Ballmann, S. Buyalo
5.5 Examples. (a) As in Example 1.1, let Y = T1 I.-JF T2 be the union of
two tori along a common face F. For this space Y, any (piecewise smooth)
nonpositively curved metric has all maximal surfaces of Y flat. On the other
hand, ~--(Y) is infinite dimensional since any two flat metrics on Tj respectively /'2 and any (piecewise smooth) locally convex isometric discs F1 C T1
respectively F2 C T2 give rise to a (piecewise smooth) nonpositively curved
metric on Y. This example demonstrates how Case 2 and Case 4 in the proof
of Theorem 5.3 occur.
(b) We now show that v(Y) may be zero without Y admitting a tight
metric. To that end, let To be a square flat torus of side length 2, and
subdivide To into four unit squares Sl . . . . . $4. Choose eight square flat tori
T b . . . , T8 of side length 2, and on each of them choose a unit square S:.
Now identify Si with S: and S~+i,
'
1 =< i < 4. Our arguments in the proof
of Theorem 1.2 show that with respect to any metric of nonpositive curvature on the resulting space Y, all the above squares will be flat with geodesic
edges. The vertices are essential, but the labelings will not be minimal: at
any such vertex v at least one interior angle of an appropriate face will be at
most ~z/2.
(c) In Example 3.13(b), i f k > 4, then va(Y) = 0 for the metric which has
all the faces regular Euclidean 2k-gons. On the other hand, va(Y) < 0 if the
faces are regular hyperbolic 2k-gons.
Now we discuss the space of tight metrics for a few classes of examples.
5.6 Theorem Let Z be a closed polyhedron whose faces are triangles and
assume that the link at any vertex of Z is metrically rigid with minimal
labeling 2zt/3. Let p : Y --~ Z be the branched covering of Z of degree 2 with
branching locus the barycenters of the faces of Z. Then
(i) any piecewise smooth metric of nonpositive curvature on Y is tight,"
(ii) for any tight metric on Y, the faces of Y are centrally symmetric flat
hexagons with angles 2~/3.
In particular, if d is a tight metric on Y and e E Ez = E is an edge of
Z, then the lengths of the two edges in Y above e coincide. Denoting this
common length by la(e), we have
(iii) the map e ~-~ la(e) defines an isomorphism between J ( Y ) and IRE+.
Note that the covering p : Y ~ Z is uniquely determined by the properties
mentioned. Indeed, restricted to the 1-skeleton, p : y(1) ~ Z(1), it corresponds
to the subgroup of rq(Z 0)) generated by the loops with an even number of
edges.
In the proof of the proposition the following lemma will be used.
5,7 Lemma Let Z' be the 2-skeleton of the 3-simplex minus an open face and
p : Y' --~ Z ~ the branched covering of Z' o f degree 2 with branching locus
the barycenters of the faces of U. Then for any geodesic metric d on Y' for
which the faces of yr are fiat hexagons with angles 2n/3, the faces of Y' are
centrally symmetric.
Metrics on 2-polyhedra
127
Fig. 7
Proof The skeleton of Y' is the graph with three faces and twelve edges as
in Fig. 7. The d-lengths of the edges satisfy a homogeneous linear system
E ' /kexp
----0;
i = 1,2,3,
k=0
where lo,
i l i~. . . . . l is are the lengths of the edges of the i-th face in consecutive order (6 real equations). This system has (real) rank 6 and hence the
assertion. []
Proof of Theorem 5.6. The faces of Y are hexagons and the links of Y are
isomorphic to those of Z under the projection p. The metric do on X for
which every face is isometric to the regular flat hexagon of side length 1 is
tight. Hence, by Theorem 3.8. any piecewise smooth metric of nonpositive
curvature on Y is tight (Y corresponding to X/F in that theorem), hence (i).
The condition on the links implies that for each vertex v of Z and each triangle
P adjacent to v there are triangles P~,P" such that PUP~UP" = Z', the complex
considered in Lemma 5.7. Hence (ii).
Obviously we can choose the lengths of the edges in Y arbitrarily, up to
the central symmetry required for each hexagon. Hence (iii).
[]
5.8 Examples. Observe that we can apply the construction of Theorem 5.6 also
to coverings of Z.
(a) Let Zn be the 2-skeleton of the (n + 1)-simplex, n > 4. Then the links
at the vertices of Z, are isomorphic to the 1-skeleton of the n-simplex, n > 4,
and hence metrically rigid. If follows from Theorem 3.8 that every geodesic
metric of nonpositive curvature of Y, is tight. The space of equivalence classes
of such metrices is isomorphic to +
by Theorem 5.6.
(b) If we choose n = 3 in the previous example, then once again by
Theorem 3.8 every geodesic metric of nonpositive curvature of ¥3 is tight.
However, the links at the vertices of ¥3 are not metrically rigid any more and
there is a choice in the angles of the hexagons. A direct computation shows that
the space of equivalence classes of metrics of g3 for which all the hexagons are
128
W. Ballmann, S. Buyalo
centrally symmetric is 15. The space J(Y3) is given as the space of solutions
of some non-linear system of equations.
(c) Let Zn be the 2-skeleton of the (n + 1)-octahedron, n > 2, and let
p : Yn ---* Zn be the covering as in Theorem 5.6. The link at any vertex of Zn
is the 1-skeleton of the n-octahedron, hence metrically rigid for n > 5 with
minimal labeling 21t/3. The polyhedron Z, has 2n(n + 1) edges, hence 9-(Yn)
is isomorphic to 11~
~,+2n(n+ 1) .
5.9 Theorem Let (X, F) be a closed 2-orbihedron such that
(i) X is a simplicial complex with every edge essential and the link of
every vertex connected and metrically rigid;
(ii) (X, F) admits a tight metric do.
Then any piecewise smooth metric of nonpositive curvature on (X, F) is
homothetic to do.
Proof By Corollary 3.9, any piecewise smooth metric of nonpositive curvature
on (X,F) is tight. By (i) and the definition of tightness, all the 2-simplices
of X are Euclidean triangles (with geodesic sides). By the rigidity of the
links, their interior angles are the same as with respect to do. Hence the
metric is homothetic to do on the triangles. Since X is connected, the claim
follows. []
5.10 Examples. (a) If X is a thick Euclidean building of dimension 2 and F
is a properly discontinuous and cocompact group of automorphisms of X, then
(X,F) satisfies the assumptions of Theorem 5.9, cf. (4.2.8).
(b) Another finite orbihedron (X,F) which is totally rigid, that is, for
every subgroup F' C F of finite index there is (up to scaling) only one
piecewise smooth nonpositively curved metric for (X, Ft), can be obtained as
follows.
The pair (X, F) will be the symmetric pair associated to the Heawood graph
A (the flag complex of the projective plane over Z/2), cf. Fig. 4, together with
the twist map z, where, for an edge e of A, z(e) is the Euclidean reflection
of the graph (as it sits in the Euclidean plane in Fig. 4) about the line perpendicular to e. As A we take the subgroup of Aut(A) generated by the "c(e),
e E EA, and we choose k(e) --- 3. Then the assumptions of Theorem 1.6.6 are
satisfied. Observe that F is cocompact and properly discontinuous since F is
transitive on the vertices of X and A is finite. Note, however, that the group
of all automorphisms of this example is not discrete, cf. [Sw]. The geodesic
metric do for which all triangles in X are equilateral Euclidean triangles is
tight.
It is interesting to consider completions or compactifications of the space
of tight metrics on a given finite 2-polyhedron or 2-orbihedron. The natural
question is whether the closure of the space of tight metrics consists of tight
metrics. We plan to come back to this question in a later publication. However,
we have the following preliminary result.
Metrics on 2-polyhedra
129
5.11 Theorem Let Y be a finite 2-polyhedron and let (d,) be a sequence o f
metrics o f nonpositive curvature on Y such that with respect to all dn and
some e > 0
(i) the shortest geodesic loop in Y has length at least 2e;
(ii) the faces of Y are flat and convex;
(iii) the edges of Y are geodesics.
Assume (dn) converges uniformly to some pseudometric d ~ on Y and let
Z = Y / ~ , where x ~ y iff d~(x, y) = O. Then Z is a finite 2-polyhedron and
the metric d on Z induced from d ~ has nonpositive curvature and satisfies
(i)-(iii).
Proof Since the projection Y ---, Z is continuous, Z is compact and Hausdorff
and hence complete with respect to d. Now Theorem 1.8 in [GLP] applies and
shows that d is geodesic.
Condition (i) implies that for any two points x , y in Y with dn(x,y) <
there is exactly one minimal d,,-geodesic connecting them. Hence the triangle
comparison holds for all dn-geodesic triangles of circumference < 2~, independently of n, see Theorem 7 in [Ba]. It follows easily that d is of nonpositive
curvature.
Condition (ii) and (iii) imply that the limit of a face in Y is either a fiat,
convex polygon or a segment or a point. Now the assertion follows. []
5.12 Remark. If ~ collapses a disjoint union of closed edges in Y to points
each and the interior angles of the faces of Y are all bounded away from n/2
(from below), independently of n, then the links of the collapsed edges are
given by the collision of the links of the ends of the corresponding edges,
see (4.3). In this case d is tight if all of these links belong to Zk or Z~, see
Proposition 4.3.6.
6 Hyperbolicity versus negative curvature
One of the starting points of our work was the question of M. Gromov whether
a word hyperbolic group admits a discrete cocompact action on some space
with a geodesic metric of negative curvature, cf. [G2]. In this section we
describe three examples of finite 2-orbihedra (X,F) with F word hyperbolic
such that (X,F) has a tight metric, but not a piecewise smooth metric of
negative curvature. Note that then also ( X , U ) does not admit a metric of
negative curvature, by Corollary 3.9, for any subgroup F' C F of finite index.
6.1 Example 1. Our first example is the "twisted case" in Example 1.5. It is
the symmetric pair (X, F ) associated to the following data:
(i) A is the 1-skeleton of the 3-simplex;
(ii) k(e) = 6 for any edge e in A;
130
W. Ballmann, S. Buyalo
(iii) for any edge e in A, the twist z(e) leaves e pointwise fixed and reflects
the edge ~ opposite to e;
(iv) A = Aut(A).
Now the geodesic metric do such that any face of X is a regular
Euclidean hexagon with side length 1 is tight. According to Proposition 1.4
and Theorem 3.8, any nonpositively curved metric for (X, F) is tight. In particular, (X, F ) does not admit a negatively curved metric.
On the other hand, F is (word) hyperbolic. Otherwise (X, do) would contain
a flat Euclidean plane E by a result of Eberlein [Eb] and Gromov [G1, p. 119].
Since F is transitive on the vertices of X, we can assume that E passes through
the preferred vertex v. Note that E is tesselated by regular Euclidean hexagons.
If f l , f z , f 3 are the hexagons in E adjacent to v and el,ez, e3 the edges in
A = Av corresponding to them, then
"c(el)(ei) = ei,
i = 2, 3 ,
by the definition of the twist map. This is a contradiction to (iii), hence X
does not contain flat Euclidean planes. Therefore F is hyperbolic.
Problem. Does this F have a torsionless subgroup of finite index? See Remark
(b) in (1.6.7).
6.2 Example 2. The second example is also a symmetric pair (X, F). The data
are as follows:
(i) A is the 1-skeleton of the 3-simplex;
(ii) k(e) = 6 for any edge e in A;
(iii) for a fixed pair e0, ~0 of opposite edges we have
~(e0)(~0) = ~o l,
z(~0)(eo) = e o l ;
for e' ~{eo,~0}, we have z(e) = ida ;
(iv) A is the group generated by the z(e), e E EA.
Now the geodesic metric do such that any face of X is a regular Euclidean
hexagon with side length 1 is tight. According to Theorem 3.8, any piecewise
smooth nonpositively curved metric for (X, F) is tight. In particular, (X, F) does
not admit a piecewise smooth negatively curved metric.
The faces of X are subdivided into two classes A and B, where A consists
of the faces with trivial twist map and B consists of the others. The definition
of • implies that each vertex v of X is adjacent to exactly four faces from
A, and they correspond to a simple loop of four edges in the link A~. Hence
the faces from A form a surface F C X which has the same 1-skeleton as
X , is tesselated by hexagons such that in each vertex four hexagons meet. In
particular, if we endow F with the length metric d such that the hexagons
are regular hyperbolic hexagons with interior angles ~/2, then F has constant
Metrics on 2-polyhedra
131
curvature - 1. The boundaries of the hexagons in X missing in F define closed
geodesics on F with respect to this metric. In particular, F is not simply
connected. The inclusion ( F , d ) ~ (X, do) is a quasi-isometry.
Note that ()(,do) cannot contain a Euclidean plane E since, for a vertex
v in E and the hexagons f l , f 2 , f 3 C E adjacent to v, the edges el,e2,e3
in At, corresponding to f l , f 2 , f 3 would constitute a simple loop of three
edges for which the transfer map would be the identity on A~. This contradicts (iii). Hence X does not contain Euclidean planes and therefore F is
hyperbolic.
6.3 Example 3. We start with a 2-polyhedron Z, whose 1-skeleton coincides
with the 1-skeleton of the product G x G, where G is the bipartite graph with
two vertices and three edges. The link of any of the four vertices is the join
A3,3 see (4.2.9). The polyhedron Z has nine faces, whose attaching map is
defined by the columns of the following (4 x 9)-matrix:
M(Z) =
[23321
1
1
2
2
2
3
3
2
2
3
3
1
2
2
3
3
2
1
3
2
1
"
Here an entry mij = k means that we attach the j-th face to the k-th edge from
the vertex i to the vertex i + 1, i mod4, where we numerate the vertices of
G x G clockwise from 1 to 4.
Let Y be the polyhedron obtained from Z by deleting the first cell. We
indicate the arguments which show that Y does not contain a flat plane. If we
also delete the fifth cell of Z, then we obtain a Klein bottle with a hole, where
the boundary of the hole is divided out by a 7Z/3-action. Hence this space does
not contain a flat plane, and hence a flat plane in Y has to pass through the fifth
cell of Z. Now there are 16 flat (3 x 3)-squares of faces in Z containing the
fifth cell. Only four of them do not contain the first cell. A direct calculation
shows that any flat (5 x 5)-extension of these also contains the first cell. Hence
Y does not contain a flat plane, and therefore Y is hyperbolic in the sense of
Gromov.
At any vertex of Y, the link is isomorphic to the join A3, 3 with one
edge deleted. Now the geodesic metric do for which all the faces are Euclidean unit squares is tight since the induced labeling on the links is minimal,
see (4.2.10). Hence Y does not admit a piecewise smooth metric of negative curvature by Theorem 3.8. On the other hand, Y does not contain Euclidean planes by what we said above. Hence the fundamental group F of Y is
hyperbolic.
Note that x(F) = x(Y) = 0, hence, by Corollary 3.12, F cannot be the
fundamental group of any simple finite polyhedron which admits a negatively
curved piecewise smooth metric.
132
W. Ballmann, S. Buyalo
7 Appendix: Piecewise smooth metrics
For a piecewise smooth metric d on a 2-polyhedron X consider the following
three conditions:
(i) d has Gauss curvature ~ K on the interior of every face;
(ii) for every edge e C X and every two faces f~, f z adjacent to e one has
kl(x) + k2(x) < 0 at any point x in the interior of e where e is smooth with
respect to f l and f2;
(iii) for every vertex v of X, the labeling ~d of Av induced by d satisfies
sys(cta) => 2~; that is, any simple loop in Av has length at least 2rt with respect
to Ad.
Here we include, by subdivision of the edge, a point x on an edge in the
set Vx of vertices if the metric d is not smooth in x with respect to some face
f adjacent to e.
7.1 Theorem Let d be a piecewise smooth metric on a 2-polyhedron X. Then
d has curvature at most K if and only if (i)-(iii) above hold
Proof Suppose a piecewise smooth metric d on X has curvature at most
K. Then clearly all the faces have Gauss curvature at most K, hence (i).
Property (iii) holds since otherwise one could find arbitrarily small digons in a
neighborhood of a point where (iii) is violated. But this would contradict the
fact that d has curvature bounded from above.
Now let xo be a point in the interior of an edge e and assume that xo is
smooth with respect to faces f l , f2 adjacent to e. Let Xl E f l , x2 E f 2 be the
points such that the geodesic segment [xlx2] in f l UeUf2 has length 2e, contains
xo as its midpoint and is orthogonal to e. Let cl, c2 be small geodesic segments
through Xl respectively x2 and perpendicular to [xlx2]. Now, for 6 > 0 small,
let x : ( - 6 , 6) ~ e be an arc length parametrization with x(0) = xo and let
L~(t) be the length of the geodesic segment between xl(t) E Cl and x2(t) E c2
which meets e perpendicularly at x(t). If e is sufficiently small, then Ls(t) is the
distance between xl(t) and x2(t) and hence L ' ( 0 ) > - O ( e ) since the metric
has an upper curvature bound. On the other hand, the second variation formula
shows L ' ( 0 ) < O(e) - kl(xo) - k2(xo), where ki(xo) is the geodesic curvature
of e at xo with respect to fi, i = 1,2. Suppose now that a piecewise smooth
metric d on X satisfies the condition (i)-(iii).
Case 1. All edges o f X are geodesics." Let x E X and let e > 0 be small. Then
any geodesic segment in the e-neighborhood Us(x) of x is minimal, and for
any point y E U~(x)\{x}, there is a unique geodesic Cy in Us(x) from x to y.
Let p ( y ) = "~ y ( o ) " E Ax. Then, for a geodesic segment c in Us(x)\{x}, p o c
is injective unless p o c is constant. Since c is minimal, the length of p o c is
less than n. If y0 denotes the initial and yl the end point of c, then (Cyo, c, @1 )
is a geodesic triangle in Us(x), see Fig. 8. N o w - o n this triangle (together
with the surface bounded by it) the metric d has curvature bounded by K
Metrics on 2-polyhedra
133
Fig. 8
by Corollary 5 in [Ba]. Hence it is thinner than the comparison triangle in the
comparison surface. Now the arguments in Sect. 3 of [Ba] apply and show that
X has curvature at most K.
Case 2. All edges of X are broken 9eodesics: This case immediately reduces
to Case 1 by subdividing X appropriately.
Case 3. General case: It suffices to consider neighborhoods of points in the
1-skeleton of X. We will approximate X locally by metric spaces with curvature
at most K, and then the assertion follows from a limit argument.
Now let x be a point in the 1-skeleton of X. Choose e > 0 small such that
U~(x) does not contain any vertices except x and such that for every edge e
going out from x, the part U,(x)N e is smooth with respect to all faces. Let
f be a face adjacent to x, bounded by the edges el and e2 going out of x.
Recall that we may think of f as embedded in an open set U of the plane
such that the boundary of f is piecewise smooth and such that the metric on
f is the restriction of a smooth Riemamlian metric on U. Now we replace the
piece U ~ ( x ) n f bounded by the original edges el and e2 by the piece bounded
by broken geodesics e~ and e~ going out of x, where the segments of e~ have
length e/n and e~' starts tangent to ei at x. Furthermore, the interior angle of
e~' at the k-th break is
k~/n
(*)
~TCk) = ~ -
f kids,
o
where ki is the geodesic curvature of ei with respect to f and s is arc length
along el.
This construction we apply simultaneously to all faces adjacent to x. Now
( , ) implies that the condition (i), (ii) and (iii) hold locally about x. Hence Case
2 applies and shows that the (local) polyhedron U~,,,(x) obtained in this way
has curvature at most K. Since U~a,(x) ~ U~(x) uniformly and any geodesic
in U~(x) is limit of geodesics in U~,n(x), the claim follows. []
134
W. Ballmann, S. Buyalo
Acknowledgement. We are grateful to F. Grunewald and G. Harder for helpful discussions.
The second author is grateful to the Sonderforschungsbereich 256 and the University of Bonn
for their hospitality during the academic year 1992/93.
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