Commensurability and quasi-isometric classification
of hyperbolic surface group amalgams
Emily Stark
November 25, 2014
Abstract. Let CS denote the class of groups isomorphic to the fundamental group
of two closed hyperbolic surfaces identified along an essential simple closed curve in
each. We construct a bi-Lipschitz map between the universal covers of these spaces
equipped with a CAT(−1) metric, proving all groups in CS are quasi-isometric. The
class CS has infinitely many abstract commensurability classes, which we characterize in terms of the ratio of the Euler characteristic of the two surfaces and the
topological type of the curves identified. We characterize the groups in CS that
contain a maximal element in their abstract commensurability class restricted to CS .
1. Introduction
To study finitely generated infinite groups, one may use algebraic and geometric classifications. Abstract commensurability defines an algebraic equivalence relation on the
class of groups, where we say two groups are abstractly commensurable if they contain
isomorphic subgroups of finite index. The geometry of finitely generated groups may be
studied by identifying a group with a family of metric spaces, which is well-defined up
to quasi-isometric equivalence and includes all Cayley graphs for the group constructed
with respect to finite generating sets. A finitely generated group is quasi-isometric to
any finite-index subgroup, so, if two finitely generated groups are abstractly commensurable, then they are quasi-isometric. Two fundamental questions in geometric group
theory are to classify the abstract commensurability and quasi-isometry classes within
a class of finitely generated groups and to understand for which classes of groups the
characterizations coincide. In this paper, we present a solution to these questions restricted to the class CS of groups isomorphic to the fundamental group of two closed
hyperbolic surfaces identified along an essential simple closed curve in each.
Our first theorem gives the quasi-isometric classification.
Theorem 1. If X1 = Sg1 ∪γ Sg10 and X2 = Sg2 ∪ρ Sg20 where Sgi and Sgi0 are closed
hyperbolic surfaces for i = 1, 2 and γ : S 1 → X1 and ρ : S 1 → X2 are the images
of essential simple closed curves under identification, then there exists a bi-Lipschitz
f1 → X
f2 mapping lifts of γ to lifts of ρ bijectively.
equivalence Φ : X
Corollary 2. If G1 , G2 ∈ CS , then G1 and G2 are quasi-isometric.
1
2
Corollary 2 also follows from Theorem 4.14 of Malone’s unpublished work in [7]; groups
in CS are examples of geometric amalgamations of free groups with the same degree
refinement, a matrix associated to the graph of groups splitting. The proof in [7]
follows the techniques of Behrstock and Neumann in [1]: a model space is constructed
by identifying the universal cover of a surface with boundary with a fattened tree
and gluing these trees along boundary curves. The bi-Lipschitz equivalence of these
spaces follows from [1], and thus these groups are shown to be quasi-isometric. In
our result, we study the CAT(−1) structure of these groups, constructing an explicit
bi-Lipschitz equivalence between the universal covers of two closed hyperbolic surfaces
identified along essential simple closed curves. The class CS is a subclass of the class
of limit groups, described in [2]; in particular, all groups in CS are constructible limit
groups of level 1. The quasi-isometric classification within the class of limit groups in
full generality remains open, and we expect the techniques of this paper may be used
towards this classification.
The class CS is also a subclass of the class C of groups isomorphic to the fundamental
group of two hyperbolic surfaces identified along a primitive closed curve in each. We
prove that within C, any group quasi-isometric to a group in CS is in CS .
Proposition 3. If G ∈ CS and G0 ∈ C\CS , then G and G0 are not quasi-isometric.
The abstract commensurability classes within CS are finer than the quasi-isometry
classes; while there is a unique quasi-isometry class in CS , there are infinitely many
abstract commensurability classes. Whyte, in [14], proves a similar result for free
products of hyperbolic surface groups, which may be thought of as fundamental groups
of closed hyperbolic surfaces identified along null-homotopic curves in each, and are
also a subclass of limit groups.
Theorem 4. ([14], Theorem 1.6, 1.7) Let Σg be the fundamental group of a surface of
genus g ≥ 2 and let m, n ≥ 2. Let Γ1 ∼
= Σa1 ∗Σa2 ∗. . .∗Σan and Γ2 ∼
= Σb1 ∗Σb2 ∗. . .∗Σbm .
Then Γ1 and Γ2 are quasi-isometric, and Γ1 and Γ2 are abstractly commensurable if
and only if
χ(Γ2 )
χ(Γ1 )
=
.
n−1
m−1
Our classification in CS up to abstract commensurability depends on Euler characteristic as well. Our techniques rely on the results of [4], [6] that the class of groups CS
and its class of finite-index subgroups are topologically rigid: any isomorphism between
subgroups of finite index in this class is induced by a homeomorphism between K(G, 1)
spaces that consist of a set of closed surfaces identified along a set of essential simple
closed curves. We prove the ratio of the Euler characteristic of the surfaces identified
and the topological type of the curves identified are obstructions to the existence of
homeomorphic finite sheeted covers, and we construct a common cover in the absence
of these obstructions. Our result is the following.
Theorem 5. If G1 , G2 ∈ CS , then G1 and G2 are abstractly commensurable if and
only if G1 and G2 may be expressed as π1 (Sg1 ) ∗ha1 i π1 (Sg10 ) and π1 (Sg2 ) ∗ha2 i π1 (Sg20 ),
3
respectively, so that ai identifies [γi ] ∈ π1 (Sgi ) and [γi0 ] ∈ π1 (Sgi0 ), and the following
conditions hold.
(a)
χ(Sg1 )
χ(Sg2 )
=
,
χ(Sg10 )
χ(Sg20 )
(b) t(γ1 ) = t(γ2 ),
(c) t(γ10 ) = t(γ20 ),
χ(S
)
r,1
where t(γ) = 1 if γ is a non-separating simple closed curve, and t(γ) = χ(Ss,1
) if γ is
a separating simple closed curve such that Sg = Sr,1 ∪γ Ss,1 and χ(Sr,1 ) ≤ χ(Ss,1 ).
The result in Theorem 5 is related to the abstract commensurability classification of
the right-angled Coxeter groups considered by Crisp and Paoluzzi in [4],
Wm,n = W (Γm,n ),
where Γm,n denotes the graph consisting of a circuit of length m + 4 and a circuit
of length n + 4 which are identified along a common subpath of edge-length 2. For
all m and n, the group Wm,n is the orbifold fundamental group of a 2-dimensional
reflection orbi-complex Om,n which is finitely covered by a space consisting of two hyperbolic surfaces identified along a single non-separating essential simple closed curve.
Conversely, all amalgams of surface groups over non-separating essential simple closed
curves are finite index subgroups of Wm,n for some m and n, dependent on the Euler
characteristic of the two surfaces. Thus, our theorem extends their result.
Corollary 6. ([4] Theorem 1.1) Let 1 ≤ m ≤ n and 1 ≤ k ≤ `. Then Wm,n and Wk,`
k
are abstractly commensurable if and only if m
n = `.
Dani and Thomas, in [5], prove a quasi-isometric classification within a class of rightangled Coxeter groups which includes Wm,n , and our result in Theorem 1 also gives
the following.
Corollary 7. ([5] Theorem 1.5) For all m, n, k, ` ≥ 1, Wm,n and Wk,` are quasiisometric.
In order to extend the result of Crisp and Paoluzzi to the setting of separating simple
closed curves, we prove the following result.
Proposition 8. Let G1 , G2 ∈ CS . If G1 and G2 are abstractly commensurable then
there exist normal subgroups N1 / G1 and N2 / G2 of finite index such that N1 ∼
= N2 .
A maximal element in the abstract commensurability class of a group G is a group
G0 that contains every other element in the abstract commensurability class of G as
a finite-index subgroup. A classic result in this setting is that of Margulis [8], who
proved that if H ≤ P SL(2, C) is a discrete subgroup of finite covolume, then there
exists a maximal element in the abstract commensurability class of H if and only if H is
non-arithmetic. It follows that the commensurability class of a non-arithmetic finitevolume hyperbolic 3-manifold contains a minimal element: there exists an orbifold
finitely covered by every other manifold in the commensurability class. Recent surveys
4
on notions of commensurability are given by Paoluzzi [10] and Walsh [13]. Within the
class CS , we prove the following analog.
Define the maximal element in the abstract commensurability class of G ∈ CS restricted
to CS to be a group G0 ∈ CS that contains every other element in the abstract commensurability class of G in CS as a finite index subgroup. We prove that a maximal
element restricted to the class CS exists only under the following conditions.
Proposition 9. Let G ∼
= π1 (Sg ) ∗hγi π1 (Sg0 ) ∈ CS be given by the monomorphisms
γ 7→ [γg ] ∈ π1 (Sg ), and γ 7→ [γg0 ] ∈ π1 (Sg0 ). Then there exists a maximal element in
the abstract commensurability class of G restricted to CS if and only if γg and γg0 are
separating simple closed curves with Sg = Sr,1 ∪γg Ss,1 , Sg0 = Sr0 ,1 ∪γg0 Ss0 ,1 and r 6= s,
r0 6= s0 .
The outline of the paper is as follows. The space and groups we consider are defined
in Section 2. In Section 3, we describe the geometry and quasi-isometry classes of
groups in CS ; we construct a bi-Lipschitz map between the universal covers of specified
K(G, 1) spaces for any G ∈ CS and prove the quasi-isometry class containing CS in C
does not contain any group not in CS . In Section 4, we describe the topology of a set
of K(G, 1) spaces and their finite-sheeted covers, prove the abstract commensurability
classification, and characterize the groups in CS that contain a maximal element in
their abstract commensurability class restricted to CS .
Acknowledgments. The author is deeply grateful for many discussions with her advisor Genevieve Walsh. The author also wishes to thank Andy Eisenberg for thoughtful
comments on a draft of this paper, Pallavi Dani for pointing out a gap in an earlier
version of Theorem 5, and her peers Jeff Carlson, Charlie Cunningham, Burns Healy,
Garret Laforge, and Chris O’Donnell for helpful conversations throughout this work.
This material is partially based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-0806676.
2. Preliminaries
An orientable surface of genus g with b boundary components will be denoted Sg,b ;
we use Sg to denote Sg,0 . A hyperbolic surface is a surface that admits a metric of
constant negative curvature. A (hyperbolic) surface group refers to the fundamental
group of a (hyperbolic) surface. A closed curve in a surface Sg,b is a continuous map
S 1 → Sg,b , and we typically identify a closed curve with its image in the surface. An
essential closed curve γ is primitive if it cannot be written as [γ] = [γ0n ] for some closed
curve γ0 . A closed curve is simple if it is embedded. A homotopy class of simple closed
curves is a homotopy class in which there exists a simple closed curve representative.
A multicurve in Sg,b is the union of a finite collection of disjoint simple closed curves
in Sg,b .
Let X denote the class of spaces homeomorphic to two hyperbolic surfaces identified
along a primitive closed curve in each. Let XS ⊂ X be the subclass in which the
5
curves that are identified are simple. Let C be the class of groups isomorphic to the
fundamental group of a space in X , and let CS ⊂ C be the subclass of groups isomorphic
to the fundamental group of a space in XS . If G ∈ C then G ∼
= π1 (Sg ) ∗hγi π1 (Sh ),
the amalgamated free product of two hyperbolic surface groups over Z. We suppress
in our notation the monomorphisms ig : hγi → π1 (Sg ) and ih : hγi → π1 (Sh ) given by
ig : γ 7→ [γg ], ih : γ 7→ [γh ], where γg : S 1 → Sg and γh : S 1 → Sh .
3. Quasi-isometric classification
3.1. The hyperbolic structure of groups in CS . Let G ∈ CS so that G ∼
= π1 (X),
1
where X = Sg ∪γ Sh ⊂ XS , and γ is the image of γg : S → Sg identified to γh : S 1 → Sh
in X. One can choose hyperbolic metrics on Sg and Sh so that the length of the
geodesic representatives of [γg ] and [γh ] is equal. Gluing by an isometry yields a
e For details on gluing
piecewise hyperbolic complex with CAT(−1) universal cover X.
e consists of convex subspaces of H2
constructions, see ([3], Section II.11.) The space X
that are the preimages of hyperbolic surfaces with boundary, identified along geodesic
e is a geometric graph
lines that are the preimages of the curve γ. The universal cover X
of spaces in the following sense. For more details on graphs of groups and graphs of
spaces, see [11], [12].
Definition 10. A geometric graph of spaces is a graph of spaces, G, consisting of a set
of vertex spaces, {Vi }i∈I , and a set of edge spaces, {Ei,j }(i,j)∈J , with J ⊂ I × I, i < j,
so that the vertex and edge spaces are geodesic metric spaces and there are isometric
embeddings
Ei,j → Vi ,
Ei,j → Vj ,
as convex subsets. The geometric realization of G is the metric space X given by the
disjoint union of the vertex and edge spaces, identified according to the relations of G,
and given the induced path metric. The underlying graph of the graph of spaces G is
the abstract graph specifying G.
To prove that all groups in CS are quasi-isometric, we exhibit a bi-Lipschitz equivalence
between the universal covers of the corresponding spaces in XS .
Definition 11. A map f : (X, dX ) → (Y, dY ) is K-bi-Lipschitz if there exists K ≥ 1
with
1
dX (x1 , x2 ) ≤ dY (f (x1 ), f (x2 )) ≤ KdX (x1 , x2 )
K
and a K-bi-Lipschitz equivalence if, in addition, it is a homeomorphism. A map is said
to be a bi-Lipschitz equivalence if it is a K-bi-Lipschitz equivalence for some K. Two
spaces X and Y are bi-Lipschitz equivalent if there exists a bi-Lipschitz equivalence
from X to Y .
We will need the following two lemmas.
6
Lemma 12. Suppose P and P 0 are compact convex hyperbolic polygons. Then P and
P 0 are bi-Lipschitz equivalent.
Lemma 13. Let T1 and T2 be tilings of convex regions R1 , R2 ⊂ H2 by compact convex
polygons of finitely many isometry types. If T1 and T2 have isomorphic dual graphs,
then R1 and R2 are bi-Lipschitz equivalent.
Proof. The tilings T1 and T2 of convex subsets of H2 are examples of locally finite
geometric graphs of spaces, where the vertex spaces are the tiles, the underlying graph
is the dual graph to the tiling, and R1 and R2 are the geometric realizations of T1
and T2 , respectively. In this language, suppose T1 and T2 consist of a set of vertex
and edge spaces {Vi }I , {Ei,j }J and {Wi }I , {Fi,j }J , respectively. Since T1 and T2 have
isomorphic dual graphs and there are finitely many isometry types of polygons, by
Lemma 12, there exists a K-bi-Lipschitz equivalence
φi : V i → Wi
for all i ∈ I such that
φi (Ei,j ) = φj (Ei,j ) = Fi,j
for all (i, j) ∈ J. There is an induced bijection
Φ : R1 → R2
defined by Φ|Vi = φi (Vi ), which is continuous by the pasting lemma.
Let x, y ∈ R1 , and let p be the geodesic path from x to y. The path p can be
n−1
decomposed into a union of geodesic segments {[xi , xi+1 ]}i=0
, with x0 = x and xn = y
so that the interior of each subpath [xi , xi+1 ] is contained entirely in a vertex space
Vi since there are finitely many isometry types of compact polygons in the tiling. If
a segment is contained in an edge space Ei,j , we may choose either Vi or Vj since
φi (Ei,j ) = φj (Ei,j ). By assumption, there exists a K-bi-Lipschitz equivalence φi :
Vi → Wi for all i. Since Φ(p) is a path connecting Φ(x) and Φ(y),
d(Φ(x), Φ(y)) =
n−1
X
d(φi (xi ), φi (xi+1 ))
i=0
≤
n−1
X
Kd(xi , xi+1 )
i=0
= K
n−1
X
d(xi , xi+1 )
i=0
= Kd(x, y).
The other inequality follows similarly. Namely, suppose q is a geodesic path from
Φ(x) to Φ(y). The path q can be decomposed into a union of geodesic segments
{[wi , wi+1 ]m−1
i=0 } where w0 = Φ(x), wm = Φ(y) and the interior of [wi , wi+1 ] is contained
7
Figure 1. On the left are lifts of a generating curve for a genus two surface,
and on the right, lifts of a curve representing the commutator element. Both
figures were drawn with Curt McMullen’s lim program [9].
entirely in a vertex space, Wi . Then, since Φ−1 (q) is a path from x to y, and φi is a
K-bi-Lipschitz equivalence for all i,
d(Φ(x), Φ(y)) =
m−1
X
d(wi , wi+1 )
i=0
≥
m−1
X
i=0
=
1
−1
d(φ−1
i (wi ), φi (wi+1 ))
K
m−1
1 X
−1
d(φ−1
i (wi ), φi (wi+1 ))
K
i=0
1
d(x, y).
≥
K
Thus, K1 d(x, y) ≤ d(Φ(x), Φ(y)) ≤ Kd(x, y), so Φ is a K-bi-Lipschitz equivalence.
e with X ∈ XS . In this section, we
3.2. Bi-Lipschitz equivalence of spaces X
e and X
e 0 with X, X 0 ∈ XS in two
construct a bi-Lipschitz equivalence between spaces X
steps. In Proposition 14 we construct a bi-Lipschitz equivalence between connected
hyperbolic regions in the complement of the set of lifts in the universal covers of X and
X 0 , giving an explicit map from the boundary lifts of one region onto the boundary lifts
in another region. Then, in Theorem 15, we extend this map to the entire universal
e and X
e 0.
covers, X
Proposition 14. Let Sg and Sh be closed hyperbolic surfaces each equipped with a
hyperbolic metric. Let γ : S 1 → Sg and ρ : S 1 → Sh be essential simple closed curves.
Let LΓ be the set of lifts of γ in H2 and let LP be the set of lifts of ρ in H2 . If Rγ is
the closure of a component of H2 \LΓ and Rρ is the closure of a component of H2 \LP ,
then there exists a bi-Lipschitz equivalence
φ̄ : Rγ → Rρ ,
mapping the lifts bounding Rγ bijectively onto the lifts bounding Rρ .
8
Figure 2. On the left is a tiling of H2 by a regular 8-gon. On the right are
the lifts of a nonseparating closed curve chosen so that each lift intersects the
fundamental domain in a single segment. Both figures were drawn with lim
[9].
Proof. Let Lγ be the set of lifts bounding Rγ and Lρ be the set of lifts bounding Rρ .
If γ̃ denotes the axis of [γ] and ρ̃ denotes the axis of [ρ], we may assume γ̃ ∈ Lγ and
ρ̃ ∈ Lρ since π1 (Sg ) and π1 (Sh ) act transitively on the set of lifts.
The outline of the proof is as follows. We first restrict to specific hyperbolic metrics on
Sg and Sh and specify a generating set S for π1 (Sg ) and a generating set T for π1 (Sh ).
We define a map between the lifts,
φ : Lγ → Lρ ,
recursively, as a union of bijective partial functions, φn . The recursion hypothesis is if
|g|S ≤ n, where |g|S denotes the word length of g with respect to the generating set S,
then gγ̃ ∈ Lγ is included in the domain of φn and if |h|T ≤ n, then hρ̃ ∈ Lρ is included
in the image of φn . Thus, the map φ is exhaustive on Lγ and Lρ .
The construction of φ induces a tiling of the regions Rγ and Rρ by compact convex
hyperbolic polygons and a homeomorphism
φ̄ : Rγ → Rρ .
Note that this tiling is determined by φ and is different from the tiling of H2 by
fundamental domains; in general, the tiling contains more than one isometry type of
tile and is not preserved by π1 (Sg ). We prove there are finitely many isometry types
of tiles in Rγ and Rρ and the dual graph to each tiling is a tree. We show φ̄ maps tiles
in Rγ to tiles in Rρ and induces an isomorphism between the dual trees. Therefore,
we may apply Lemma 13 to conclude that Rγ and Rρ are bi-Lipschitz equivalent.
To make these notions precise, we begin by restricting to specific hyperbolic metrics on
Sg and Sh and giving the non-standard generating sets S and T , defined with respect
9
h
g
Figure 3. The generating set S is defined so that dS (g, h) = 1 since g and
h lie in fundamental domains that share a vertex.
to these metrics and chosen to make the recursion hypotheses hold. The following,
when described for γ and π1 (Sg ), holds likewise for ρ and π1 (Sh ).
Any two hyperbolic metrics on Sg are bi-Lipschitz equivalent, so assume the hyperbolic
metric on Sg is given by a representation π1 (Sg ) → P SL(2, R) with a fundamental
domain for the action of π1 (Sg ) on the hyperbolic plane a 4g-gon and so that the axis
of [γ], γ̃ : R → H2 , intersects each translate of the fundamental domain in a single
geodesic segment. Parametrize γ̃ so that γ̃(n) is the midpoint of a geodesic segment
in a translate of the fundamental domain for all n ∈ Z. Since π1 (Sg ) acts on H2 by
isometries, gγ̃(n) is the midpoint of a geodesic segment of gγ̃ for all g ∈ π1 (Sg ) and all
n ∈ Z. Give the set of lifts a cell structure by declaring gγ̃(n) a 0-cell for all g ∈ π1 (Sg )
and n ∈ Z.
Let S be the generating set for π1 (Sg ) that consists of the 16g 2 − 8g elements of π1 (Sg )
that map γ̃(0) onto the fundamental domains that share a vertex with the fundamental
domain containing γ̃(0). Let dS be the word metric on π1 (Sg ) with generating set S.
Since there is a one-to-one correspondence between 0-cells in H2 and elements of π1 (Sg ),
define dS (gγ̃(s), g0 γ̃(r)) to be dS (gγ s , g0 γ r ).
The map between the set of lifts must preserve the ordering on the boundary, so orient
∂∞ H2 , the visual boundary of H2 . View ∂∞ H2 ∼
= S 1 as the unit interval [0, 1] with the
identification 0 ∼ 1. A geodesic line, `(t), defines two points on the boundary, which
we denote `+∞ and `−∞ . Orient the boundary clockwise so that γ̃ +∞ = 0. We may
assume that for all lifts gγ̃ ∈ Lγ , ∂∞ (gγ̃) ⊂ S 1 \[0, γ̃ −∞ ]. Since γ and ρ are simple
closed curves, the order on the boundary gives a total order on the lifts in Lγ and Lρ ;
we say gγ̃ < hγ̃ if ∂∞ (gγ̃) < ∂∞ (hγ̃).
Construction:
In this section, we specify a map between the lifts, φ : Lγ → Lρ , and a map between
S
S
the regions φ̄ : Rγ → Rρ , recursively, with φ = n φn and φ̄ = n φ̄n . The maps φn
10
induce a tiling of Rγ and Rρ by compact convex hyperbolic polygons and the map
φ̄n : Rγ → Rρ . The recursion hypotheses are:
(1) If |g|S ≤ n, then gγ̃ ∈ Lγ is included in the domain of φn , and if |h|T ≤ n, then
hρ̃ ∈ Lρ is included in the image of φn .
(2) Each tile has at most M = 3 + max{64g 2 − 32g, 64h2 − 32h} edges, and each
edge constructed in the interior of Rγ connects vertices g0 γ̃(r) and gγ̃(s) with
dS (g0 γ̃(r), gγ̃(s)) = 1.
(3) The dual graph to each tiling is a tree, and φ̄ maps tiles in Rγ to tiles in Rρ ,
inducing an isomorphism between the dual trees.
Stage one:
Begin by choosing the set of lifts of γ and ρ in the domain and image of φ1 . Let Wγ1
and Wρ1 be the set of all vertices in Rγ and Rρ at distance 1 from γ̃(0) and ρ̃(0) in
the metrics dS and dT , respectively, and select and label every lift that intersects these
sets as follows. No lift intersects two edges of any fundamental domain that share a
vertex, so a lift intersects Wγ1 in at most three vertices, {gγ̃(k − 1), gγ̃(k), gγ̃(k + 1)}
for some k ∈ Z, as illustrated in Figure 4.
Let
Sγ1 = {g1 γ̃(0), . . . , gm γ̃(0)} ⊂ Wγ1
Sρ1 = {h1 ρ̃(0), . . . , hn ρ̃(0)} ⊂ Wρ1
be the subset of vertices of Wγ1 and Wρ1 so that for Sγ1 :
(a) One vertex of each lift that intersects Wγ1 is included.
(b) If a lift intersects Wγ1 in three vertices, the middle vertex is chosen.
(c) The vertices are labeled so that gi γ̃ < gj γ̃ if i < j.
Likewise for Sρ1 . Condition (a) ensures recursion hypothesis (1) is satisfied, and condition (b) ensures dS (gi γ̃(0), gi+1 γ̃(0)) = 1 and dT (hi ρ̃(0), hi+1 ρ̃(0)) = 1, and condition
(c) is used to preserve the ordering on the boundary.
If m = n, continue to the definition of φ1 . Otherwise, if m < n, let Wγ1,r be the set of
all vertices in Rγ at distance 1 from {γ̃(1), γ̃(2), . . . , γ̃(r)}, and select the n − m lifts
that intersect Wγ1,r lowest in the ordering from the boundary. If there do not exist
n − m such lifts that intersect Wγ1,r , increase r. Choose and label one vertex of each of
these lifts so that if a lift intersects Wγ1,r in three vertices, the middle vertex is chosen,
and the selected vertices are labeled gm+1 γ̃(0), . . . , gn γ̃(0) so that gi γ̃ < gj γ̃ if i < j.
Then dS (gi γ̃(0), gi+1 γ̃(0)) = 1 for 1 ≤ i < n.
11
~
γ(0)
Rγ
~
γ(1)
~
γ(3)
~
γ(2)
Figure 4. A sketch of the lift γ̃, the intersection of Rγ and the fundamental
domains γ̃ passes through, and the vertices at distance 1 from the vertices
labeled. The small arrow indicates the orientation on ∂∞ H2 .
We define φ1 : gi γ̃ 7→ hi ρ̃; to give the map explicitly, preserving the orientation on the
boundary, re-label the lifts by
L1γ = {`eg (t), `g1 (t), . . . , `gn (t)} ⊂ Lγ
L1ρ = {`eh (t), `h1 (t), . . . , `hn (t)} ⊂ Lρ
where `gi = gi γ̃, eg and eh denote the identity elements, and `gi is parametrized so
that `gi (0) = gi γ̃(0), dS (`gi (p), `gi (q)) = |p − q| for p, q ∈ Z, and `+∞
< `−∞
gi
gi ; likewise
1
for Lρ . Define
φ1 : L1γ → L1ρ
by,
φ1 (`eg (t)) = `eh (t),
φ1 (`gi (t)) = `hi (t)
for all 1 ≤ i ≤ n.
If m ≤ n, construct a hyperbolic polygon, Pγ1 , in Rγ with vertices
Vγ1 = {γ̃(−1), g1 γ̃(0), . . . , gn γ̃(0), γ̃(r)},
edges connecting vertices adjacent in the list, and an edge connecting γ̃(r) and γ̃(−1),
where r is given above if m < n and r = 1 if m = n. Similarly, construct a hyperbolic
polygon, Pρ1 , in Rρ with vertices
Vρ1 = {ρ̃(−1), h1 ρ̃(0), . . . , hn ρ̃(0), ρ̃(1)},
edges connecting vertices adjacent in the list, and an edge connecting ρ̃(1) and ρ̃(−1).
If m > n, the process is carried out in reverse, and the final vertices in the sets Vγ1 and
Vρ1 are γ̃(1) and ρ̃(s) for some s. Define
φ̄1 : Pγ1 7→ Pρ1
12
Figure 5. A sketch of the lifts mapped during stage 1, and, below, the first
two polygons constructed in the tilings of Rγ and Rρ .
by mapping the vertices in Vγ1 to the vertices in Vρ1 , in order, and extending to a
homeomorphism on the edges and interior.
The polygons Pγ1 and Pρ1 have an equal number of sides, M , by construction. We claim
3 ≤ M ≤ 2 + max{16g 2 − 8g, 16h2 − 8h}.
By the choice of representation, each translate of the fundamental domain that intersects Rγ intersects Rγ in at least two vertices, so m, n ≥ 1, proving the lower bound.
On the other hand, m, n ≤ max{16g 2 − 8g, 16h2 − 8h} by the choice of generating
sets, proving the upper bound. In addition, each edge constructed in the interior of
Rγ satisfies dS (gi γ̃(0), gi+1 γ̃(0)) = 1 for 1 ≤ i ≤ n − 1. Likewise for Pρ1 . The dual tree
after stage one is a single point, thus, all recursion hypotheses are satisfied.
Stage n:
Suppose φn−1 , n − 1 stages of the tiling, and φ̄n−1 have been constructed, satisfying
the hypothesis of the recursion. For each of the K edges constructed in the interior of
Rγ during stage n − 1, extend the maps φn−1 and φ̄n−1 and the tiling as follows.
Suppose {g0 γ̃(0), gγ̃(0)} is the ith edge constructed during stage n − 1 in the interior
of Rγ , g0 γ̃ < gγ̃, and φn−1 induces the map φ̄n−1 : {g0 γ̃(0), gγ̃(0)} 7→ {h0 ρ̃(0), hρ̃(0)}.
13
Figure 6. A sketch of the sets (1) and (2) and the fundamental domains
that contain these elements, which are drawn in dashed lines. The shaded
regions indicate tiles constructed during previous stages.
Consider
(1)
{g0 γ̃(−1), g0 γ̃(0), gγ̃(0), gγ̃(1)},
(2)
{h0 ρ̃(−1), h0 ρ̃(0), hρ̃(0), hρ̃(1)}.
In a manner similar to that of stage 1, first choose the lifts of γ and ρ in the domain
and image of φni . Let Wγni and Wρni be the set of all vertices in Rγ and Rρ at distance
1 from (1) and (2) that do not lie on a lift included in the domain or image of φn−1 ,
respectively. Select and label each lift that intersects one of these sets as follows. Let
Sγni = {gi,1 γ̃(0), . . . , gi,r γ̃(0)} ⊂ Wγni
Sρni = {hi,1 ρ̃(0), . . . , hi,s ρ̃(0)} ⊂ Wρni
be the subset of vertices of Wγni and Wρni chosen so that for Sγni :
(a) One vertex of each lift that intersects Wγni is included.
(b) If a lift intersects Wγni in three vertices, the middle vertex is chosen.
(c) The vertices are labeled so that gi γ̃ < gj γ̃ if i < j.
Likewise for the vertices in Sρni . Condition (b) ensures dS (gi,k γ̃(0), gi,k+1 γ̃(0)) = 1 and
dT (hi,k ρ̃(0), hi,k+1 ρ̃(0)) = 1.
If r = s, continue to the definition of φni . Otherwise, if r < s, let Wγni ,k be the set
of all vertices in Rγ at distance 1 from {gγ̃(2), gγ̃(3), . . . , gγ̃(k)} that do not lie on
a lift included in the domain of φn−1 , and select the s − r lifts that intersect Wγni ,k
lowest in the ordering from the boundary. If there do not exist s − r such lifts, increase
k. Choose and label one vertex of each of these lifts so that if a lift intersects Wγni ,k
14
Figure 7. An illustration of the new lifts that are mapped at stage n, and
below, tiles constructed during stage n.
in three vertices, the middle vertex is chosen, and the vertices selected are labeled
gi,r+1 γ̃(0), . . . , gi,s γ̃(0) so that gi,j γ̃ < gi,k γ̃ if j < k. Then dS (gi,j γ̃(0), gi,j+1 γ̃(0)) = 1
for 1 ≤ j < s.
As before, we map gi,j γ̃ to hi,j ρ̃; to define the map φni , label these lifts
Lnγ i = {`gi,1 (t), . . . , `gi,s (t)} ⊂ Lγ ,
Lnρ i = {`hi,1 (t), . . . , `hi,s (t)} ⊂ Lρ
where `gi,k = gi,k γ̃ and is parametrized so that `gi,k (0) = gi,k γ̃(0), dS (`gi,k (p), `gi,k (q)) =
−∞
ni
|p − q| for p, q ∈ Z and `+∞
gi,k < `gi,k ; likewise for Lρ . Define
φni : Lnγ i → Lnρ i
by
φni (`gi,j (t)) = `hi,j (t)
for all 1 ≤ j ≤ s and let
φn = φn−1 ∪
K
[
φni .
i=1
If r ≤ s, construct a hyperbolic polygon, Pγni in Rγ with vertices
Vγni = {g0 γ̃(0), g0 γ̃(−1), gi,1 γ̃(0), . . . , gi,s γ̃(0), gγ̃(k), gγ̃(1), gγ̃(0)},
15
edges connecting adjacent vertices in the list, and an edge connecting g0 γ̃(0) and gγ̃(0),
where k is given above if r < s and k = 2 if r = s. Construct a hyperbolic polygon,
Pρni in Rρ with vertices
Vρni = {h0 ρ̃(0), h0 ρ̃(−1), hi,1 ρ̃(0), . . . , hi,s ρ̃(0), hρ̃(2), hρ̃(1), hρ̃(0)},
edges connecting vertices adjacent in the list, and an edge connecting h0 ρ̃(0) and hρ̃(0).
If r > s, the process is carried out in reverse. Define
φ̄ni : Pγni 7→ Pρni
by mapping vertices in Vγni to vertices in Vρni in the order given in the lists, and
extending to the edges and interior by a homeomorphism. Let
φ̄n = φ̄n−1 ∪
K
[
φ̄ni .
i=1
To prove the first recursion hypothesis holds after stage n, consider Γ, the Cayley
graph for π1 (Sg ) with generating set S, which maps into the hyperbolic plane with the
0-cells, {gγ̃(0)}, previously described and edges that are geodesic segments. A more
standard Cayley graph, Γ0 , for π1 (Sg ) maps into H2 dual to the tiling by 4g-gons; there
is a 4g-gon in Γ0 about each vertex in the tiling. The Cayley graph Γ is obtained from
the Cayley graph Γ0 by replacing each 4g-gon with K4g , the complete graph on 4g
vertices. Let ΓR be the subgraph of Γ induced by vertices contained in Rγ . That is,
ΓR consists of all vertices that lie on the lifts in Lγ and all the edges in Γ between these
vertices. The subgraph ΓR is connected since any fundamental domain that intersects
Rγ intersects Rγ in at least two vertices. Now, let gγ̃ ∈ Rγ with |g|S = n. Since
the subgraph ΓR is connected, there exists g 0 ∈ ΓR with dS (gγ̃(0), g 0 γ̃(0)) = 1 and
|g 0 |S ≤ n − 1. By the recursion hypothesis, g 0 γ̃ is in the domain of φn−1 , so g 0 γ̃(0) lies
on the edge of a polygon in the interior of Rγ . Hence gγ̃ is in the domain of φn by the
choice of elements in the set (1) and Wγni ,k .
The bound on the number of sides of the polygons Pγni and Pρni holds by the choice of
elements in Wγni ,1 , Wρni ,1 and the size of the generating sets, which imply
(3)
r ≤ 64g 2 − 32g and s ≤ 64h2 − 32h.
Each edge of Pγni in the interior of Rγ satisfies
dS (gi,j γ̃(0), gi,j+1 γ̃(0)) = 1, dS (hi,j ρ̃(0), hi,j+1 ρ̃(0)) = 1
for 1 ≤ j ≤ n − 1 by construction.
Finally, the dual tree associated to φ̄ni has one additional edge, connecting the tile
constructed in stage n−1 containing g0 γ̃(0) and gγ̃(0) to the tile constructed associated
to φni . Since φ̄ still maps the edge {g0 γ̃(0), gγ̃(0)} to the edge {h0 ρ̃(0), hρ̃(0)}, φ̄
extends the isomorphism between the dual trees.
16
Finitely many isometry types of tiles:
By this construction, Rγ and Rρ are tiled by compact convex hyperbolic polygons
and φ induces an isomorphism between the dual trees of the tilings. The final claim
needed to conclude the proof is that there are only finitely many isometry types of
tiles constructed.
By the recursion hypothesis, each polygon in Rγ has at most M edges, where M =
3 + max{64g 2 − 32g, 64h2 − 32}. To bound the edge lengths, observe that all but
possibly one edge of the polygon connects vertices v and w with dS (v, w) = 1. There
are at most 16g 2 − 8g edge lengths of these edges; let L be the length of the longest.
The last edge connects gγ̃(k) and gγ̃(1) and k ≤ 64h2 − 32h. So the maximum length
of this edge is k × d, where d is the translation length of [γ]. Let N = max{L, n × d}.
Let P be a polygon constructed in the tiling and fix a vertex gγ̃(0) in the polygon.
All other vertices lie in Bgγ̃(0) (r), a ball of radius r = M × N in the hyperbolic
plane with center gγ̃(0). The element g −1 ∈ π1 (Sg ) is an isometry mapping gγ̃(0)
to γ̃(0), and Bgγ̃(0) (r) to Bγ̃(0) (r), a ball of radius r about γ̃(0). Since the group
action preserves the 0-cells, g −1 maps P to a polygon in Bγ̃(0) (r) with vertices gi γ̃(0)
and dS (γ̃(0), gi γ̃(0)) ≤ r. There are only finitely many polygons in this ball with
vertices gi γ̃(0), so there are only finitely many isometry types of polygons constructed
in the tiling. A similar argument holds for the tiling of Rρ , concluding the proof of
Proposition 14.
To prove the entire universal covers of X1 , X2 ∈ XS are bi-Lipschitz equivalent, we map
the regions incident to the axis of [γ] to the regions incident to the axis of [ρ] using the
bi-Lipschitz equivalence from Proposition 14. We use the action of the groups by deck
transformations on the universal covers to extend the map recursively along regions
incident to the lifts that bound a region already in the domain.
Theorem 15. If X1 = Sg1 ∪γ Sg10 and X2 = Sg2 ∪ρ Sg20 where Sgi and Sgi0 are closed
hyperbolic surfaces for i = 1, 2 and γ : S 1 → X1 and ρ : S 1 → X2 are the images
of essential simple closed curves under identification, then there exists a bi-Lipschitz
f1 → X
f2 mapping lifts of γ to lifts of ρ bijectively.
equivalence Φ : X
Proof. Let γ̃ and ρ̃ be the axes for [γ] and [ρ], respectively. Then γ̃ is contained in
a hyperbolic plane stabilized by π1 (Sg1 ) and a hyperbolic plane stabilized by π1 (Sg10 ),
and ρ̃ is contained in a hyperbolic plane stabilized by π1 (Sg2 ) and a hyperbolic plane
stabilized by π1 (Sg20 ). Let R1 and R2 be the convex regions bounded by lifts incident to
γ̃ in the plane stabilized by π1 (Sg1 ) and let R3 and R4 be the convex regions bounded
by lifts incident to γ̃ in the plane stabilized by π1 (Sg10 ). Similarly, let S1 and S2 be the
convex regions bounded by lifts incident to ρ̃ in the plane stabilized by π1 (Sg2 ) and let
S3 and S4 be the convex regions bounded by lifts incident to ρ̃ in the plane stabilized
by π1 (Sg20 ).
17
By Proposition 14, there exist Ki -bi-Lipschitz equivalences
φ̄i : Ri → Si
that are bijective on the lifts bounding Ri and Si and each map γ̃ to ρ̃. Let K =
max{Ki } so that φi is a K-bi-Lipschitz equivalence for 1 ≤ i ≤ 4. Define
Φ|Ri = φ̄i .
To extend the map coherently along the regions incident to a lift already included
f1 and X
f2 with colors
in the domain, color the convex regions bounded by lifts in X
{C1 , C2 , C3 , C4 } as follows. Let C(R) denote the color of region R. Two-color the
plane stabilized by π1 (Sg1 ) with C1 and C2 so that C(Ri ) = Ci and so that adjacent
regions have distinct colors. Do the same for the planes stabilized by π1 (Sg10 ), π1 (Sg2 ),
f1 and
and π1 (Sg0 ) so that C(Ri ) = C(Si ) = Ci . Extend the coloring to a coloring of X
2
f2 recursively as follows. Suppose a hyperbolic plane, H, stabilized by a conjugate of
X
π1 (Sg1 ) or π1 (Sg10 ) has not been colored and that H intersects, in a lift gγ̃, a hyperbolic
plane that has been colored. Then g −1 H contains γ̃, and has been colored. For every
f2 in analogously.
region R ⊂ H, let C(R) = C(g −1 R). Color X
Observe that, regardless of the topological type of γi and γi0 , π1 (Xi ) is transitive on
f1 and X
f2 .
the regions restricted to one color in X
Define the map Φ recursively. Suppose R is a region that intersects, in a lift gγ̃, the
f2 . Then Φ : gγ̃ 7→ g 0 ρ̃ for some
boundary of a region already mapped by Φ to X
f2 intersecting g 0 ρ̃ with C(R) = C(S) = Ci for
g 0 ∈ π1 (X2 ). Let S be the region in X
some i, 1 ≤ i ≤ 4. Then, map R to S by the map
Φ|R = g 0 ◦ φi ◦ g −1 ,
a K-bi-Lipschitz equivalence since g and g 0 are isometries.
3.3. Amalgams over immersed curves. All groups in CS are quasi-isometric, and
the quasi-isometric classification of groups of the form π1 (Sg ) ∗hai π1 (Sh ) in full generality depends on the image of a in the groups π1 (Sg ) and π1 (Sh ).
Proposition 16. If G ∈ CS and G0 ∈ C\CS , then G and G0 are not quasi-isometric.
Proof. All groups in C are δ-hyperbolic, so any quasi-isometry between groups in this
class induces a homeomorphism between their visual boundaries. The boundary of a
lift of the curves identified is the intersection of two circles in the boundary of the group,
∂G. The number of components in the complement of the boundary of a lift depends
on whether the curves identified have simple representatives on the surface. That is,
if both curves identified are simple, there are four components in the complement of
the boundary of a lift in ∂G, while if at least one curve has no simple representative,
there are fewer than four components in the complement of the boundary of a lift in
∂G. A homeomorphism preserves the number of components in the complement of the
boundary of a lift, so the fundamental group of a space obtained by identifying closed
18
hyperbolic surfaces along two essential simple closed curves is not quasi-isometric to
the fundamental group of a space obtained by identifying along at least one curve with
no simple representative.
Corollary 17. If G ∈ CS and G0 ∈ C\CS , then G and G0 are not abstractly commensurable.
The quasi-isometric and abstract commensurability classification of groups in C remains open.
4. Abstract commensurability classification
In this section, we give the abstract commensurability classification for groups in CS
and characterize the groups in CS that contain a maximal element in their abstract
commensurability class restricted to CS . We use `cm(a, b) to denote the least common
multiple of a and b and gcd(a, b) to denote the greatest common divisor of a and b.
4.1. The topology of spaces in XS . The subgroup structure of an amalgamated
product is described in the following theorem of Scott and Wall.
Theorem 18. ([11], Theorem 3.7) If G ∼
= A ∗C B and if H ≤ G, then H is the
fundamental group of a graph of groups, where the vertex groups are subgroups of
conjugates of A or B and the edge groups are subgroups of conjugates of C.
Topologically, Theorem 18 implies that any finite-sheeted cover of the space X =
Sg ∪γ Sh , where γ is the image of γg : S 1 → Sg and γh : S 1 → Sh under identification,
consists of a set of surfaces which cover Sg and a set of surfaces which cover Sh ,
identified along multicurves that are the preimages of γg and γh . These covers are
examples of simple, thick, 2-dimensional hyperbolic P-manifolds (see [6], Definition
2.3.) So, the following topological rigidity theorem of Lafont allows us to address the
abstract commensurability classification for members in CS from a topological point of
view. Corollary 20 also follows from the proof of Proposition 3.1 in [4].
Theorem 19. ([6], Theorem 1.2) Let X1 and X2 be a pair of simple, thick, 2dimensional hyperbolic P -manifolds, and assume that φ : π1 (X1 ) → π1 (X2 ) is an
isomorphism. Then there exists a homeomorphism Φ : X1 → X2 that induces φ on the
level of fundamental groups.
∼ π1 (X), G0 =
∼ π1 (X 0 ) and X, X 0 ∈ XS . Then
Corollary 20. Let G, G0 ∈ CS with G =
G and G0 are abstractly commensurable if and only if X and X 0 have homeomorphic
finite-sheeted covering spaces.
In the proof of the classification theorem, we make use of the corollary of the following
proposition, which, topologically, implies that if X, X 0 ∈ XS and π1 (X) and π1 (X 0 ) are
abstractly commensurable, then X and X 0 have homeomorphic finite-sheeted regular
covering spaces.
19
Proposition 21. Let G1 , G2 ∈ CS . If G1 and G2 are abstractly commensurable, then
there exist normal subgroups of finite index, N1 / G1 and N2 / G2 , such that N1 ∼
= N2 .
Proof. Suppose G1 and G2 are abstractly commensurable. Let X1 , X2 ∈ XS with
G1 ∼
= π1 (X2 ). By Corollary 20, there exist finite-sheeted covering
= π1 (X1 ) and G2 ∼
spaces Y1 → X1 and Y2 → X2 that are homeomorphic, Y1 ∼
= Y2 .
The spaces X1 and X2 may be subdivided into finite simplicial complexes, which lifts to
a realization of Y1 and Y2 as finite simplicial complexes. There is a common refinement
of these simplicial realizations, and a simplicial homeomorphism Y1 → Y2 .
Let X denote the universal cover of Y1 and Y2 , equipped with the simplicial structure.
Let Aut(X) denote the group of simplicial automorphisms of X. Then G1 , G2 ≤
Aut(X). We claim that these are subgroups of finite index; we will show X/G1 is a
finite-sheeted cover of X/Aut(X), and likewise for X/G2 . To see this, observe that a
simplex modulo its automorphism group has positive area so X/Aut(X) has positive
area. Since X/G1 has finite area, it forms a finite-sheeted cover of X/Aut(X); likewise
for X/G2 .
So, G1 , G2 ≤\
Aut(X) are finite-index subgroups, hence so is G1 ∩ G2 ≤ Aut(X). The
subgroup
g(G1 ∩ G2 )g −1 is then a normal finite index subgroup of both G1
g∈Aut(X)
and G2 .
To study finite-sheeted covering spaces, we use the following well-known fact.
Lemma 22. If X is a CW-complex and X 0 is a degree n cover of X, then χ(X 0 ) =
nχ(X), where χ denotes Euler characteristic.
A converse to Lemma 22 holds for hyperbolic surfaces with one boundary component,
which we use to construct covers of spaces in XS .
Lemma 23. For gi ≥ 1, if χ(Sg2 ,1 ) = nχ(Sg1 ,1 ), then Sg2 ,1 n-fold covers Sg1 ,1 .
Proof. Let
π1 (Sgi ,1 ) = ha1 , b1 , . . . , agi , bgi | i ∼
=F
2gi
be presentations for the fundamental groups of Sgi ,1 . Then, the homotopy class of
the boundary element γi : S 1 → Sgi ,1 corresponds to the element [a1 , b1 ] . . . [agi , bgi ] ∈
π1 (Sgi ,1 ).
We exhibit π1 (Sg2 ,1 ) as an index n subgroup of π1 (Sg1 ,1 ) so that in the corresponding
cover, γ2 has preimage a single curve that n-fold covers γ1 .
Realize π1 (Sg1 ,1 ) as the fundamental group of a wedge of 2g1 oriented circles labeled
by the generating set. Construct an n-fold cover of this space as a graph, Γ, on n
vertices labeled {0, . . . , n − 1}. For every generator besides a1 , construct an oriented
n-cycle on the n vertices with each edge labeled by the generator. Since χ(Sg1 ,1 ) and
20
a1
b1
2
b1
a2
a2
1
b2
a1
b2
3
a2
b1
b2
a1 b1 a2 b2 a1
0
b2
4
b2
a2
b1
a1
a2
b1
a1
b2
6
a2
5
b1
a1
a2
a1
b1
b2
Figure 8. A covering map that realizes S11,1 as a 7-fold cover of S2,1 .
χ(Sg2 ,1 ) are both odd, n must be odd as well. Let {i, i + 1} and {i + 1, 1} be directed
edges labeled by a1 for i < n and i odd. Construct a directed loop labeled a1 at vertex
{0}. By construction, Γ covers the wedge of circles given above.
To see that γ1 has a preimage with one component, choose a vertex v in the graph Γ
and consider the edge path p with edges labeled ([a1 , b1 ] . . . [ag1 , bg1 ])k , a preimage of
a representative of γ1 . Then n is the smallest non-zero k for which p terminates at v.
To see this, note that it suffices to consider the path p0 = [a1 , b1 ]k since every other
segment [aj , bj ] returns to its initial vertex. Starting at vertex {0}, observe that the
path [a1 , b1 ]k terminates at the vertex labeled



2k − 1
2n − 2k


0
proving the claim.
if 0 < k < b n2 c mod n
if b n2 c ≤ k < n mod n
if k = 0 mod n,
21
If γ : S 1 → Sg is an essential simple closed curve, then up to a homeomorphism of the
surface Sg , the curve γ is either non-separating or separating so that Sg = Sr,1 ∪γ Ss,1 .
We characterize the topological type of γ with the following notation.
Definition 24. Let γ : S 1 → Sg be an essential simple closed curve. Let t(γ), the
χ(Sr,1 )
topological type of γ, be 1 if γ is non-separating and χ(Ss,1
) , where χ(Sr,1 ) ≤ χ(Ss,1 ), if
γ is separating.
Definition 25. If Sg and Sh are closed hyperbolic surfaces, γ is a multicurve on Sg
and ρ is a multicurve on Sh , we say (Sg , γ) covers (Sh , ρ) if there exists a covering map
p : Sg → Sh so that γ is the full preimage of ρ in Sg , γ = p−1 (ρ).
Proposition 26. Let Sg1 and Sg2 be closed hyperbolic surfaces and γ1 : S 1 → Sg1 and
γ2 : S 1 → Sg2 essential simple closed curves. There exists (Sg , γ) which covers both
(Sg1 , γ1 ) and (Sg2 , γ2 ) if and only if t(γ1 ) = t(γ2 ).
Proof. Suppose there exists (Sg0 , γ0 ) which covers both (Sg1 , γ1 ) and (Sg2 , γ2 ). Then
there exists (Sg , γ) a regular cover of (Sg1 , γ1 ) and a cover of (Sg2 , γ2 ).
First suppose that γ1 is a non-separating simple closed curve; we claim that t(γ2 ) = 1.
Either γ2 is non-separating, or γ2 is separating so that Sg2 = Sr,1 ∪γ2 Ss,1 . Since the
cover of Sg1 is regular, Sg \γ consists of a set of homeomorphic surfaces with boundary.
Color Sr,1 red and Ss,1 blue, which lifts to a coloring of the components of Sg \γ. Let
CR denote the red components of Sg \γ and let CS denote the blue components of
Sg \γ; then, CR covers Sr,1 and CS covers Ss,1 . The colored boundary components of
the surfaces in Sg \γ come in red-blue pairs that project to γ2 under the covering map.
Since all components of Sg \γ are homeomorphic, there are an equal number of red and
blue components. Thus, χ(CR ) = χ(CS ). So,
χ(CS ) = χ(CR ) = nχ(Sr,1 ) = nχ(Ss,1 ),
and therefore r = s, and t(γ2 ) = 1. Likewise, if γ2 is non-separating, then t(γ1 ) = 1.
Otherwise, both γ1 and γ2 are separating simple closed curves with Sg1 = Sr1 ,1 ∪γ1 Ss1 ,1
and Sg2 = Sr2 ,1 ∪γ2 Ss2 ,1 , and ri ≤ si . Since the cover of Sg1 is regular, Sg \γ consists
of two homeomorphism classes of surfaces with boundary: Sgr ,br which cover Sr1 ,1 and
Sgs ,bs which cover Ss1 ,1 . Since there is a homeomorphism of Sg taking any lift of γ2 to
any other lift of γ2 , we may assume Sgr ,br cover Sr2 ,1 and Sgs ,bs cover Ss2 ,1 , as well.
Suppose that Sgr ,br forms a ki degree cover of Sri ,1 and that Sgs ,bs forms an `i degree
cover of Ssi ,1 . Further, suppose that Sg contains K subsurfaces homeomorphic to Sgr ,br
and L subsurfaces homeomorphic to Sgs ,bs . If Sg is an ni degree cover of Sgi , then
(4)
ni = L`i = Kki .
By Lemma 22,
(5)
χ(Sgr ,br ) = k1 χ(Sr1 ,1 ) = k2 χ(Sr2 ,1 ),
22
χ(Sgs ,bs ) = `1 χ(Ss1 ,1 ) = `2 χ(Ss2 ,1 ).
Combining (4) and (5), we have
χ(Sr1 ,1 )
χ(Sr2 ,1 )
=
,
χ(Ss1 ,1 )
χ(Ss2 ,1 )
so t(γ1 ) = t(γ2 ).
For the converse, let L = −`cm(|χ(Sg1 )|, |χ(Sg2 )|) and let ki = χ(SLg ) . We construct
i
(Sg , γ) so that χ(Sg ) = 2L and γ has two components, each of which covers γi by
degree ki .
First suppose that t(γ1 ) = t(γ2 ) = 1. Let Sg be the closed surface with Euler characteristic 2L. Let γ be a multicurve on Sg so that Sg \γ consists of two surfaces with
Euler characteristic L and two boundary components. We prove (Sg , γ) covers both
(Sg1 , γ1 ); the proof that (Sg , γ covers (Sg2 , γ2 ) is similar. If γ1 is non-separating, the
pair (Sg , γ) covers (Sg1 , γ1 ) via an intermediate cover
2
k
1
(Sg , γ) −→ (Sf
g1 , γe1 ) −→ (Sg1 , γ1 ).
Let Sf
g1 be the closed surface with Euler characteristic L. Arranging the handles
symmetrically about a central handle, choose γe1 : S 1 → Sf
g1 a non-separating simple
2π
closed curve so that upon rotation by k1 , γe1 projects to γ1 and (Sf
g1 , γe1 ) is a cyclic
f
cover of (Sg , γ1 ) of degree k1 . Then (Sg , γ) covers (Sg , γe1 ): arrange the handles
1
1
symmetrically about the multicurve γ and rotate by π.
Otherwise, γ1 is separating with Sg1 = Sr,1 ∪γ1 Ss,1 and r = s. If g2 is even, the pair
(Sg , γ) covers (Sg1 , γ1 ) via an intermediate cover
2
k
1
(Sg , γ) −→ (Sf
g1 , γe1 ) −→ (Sg1 , γ1 ),
Since g2 is even, L is congruent to 2 modulo 4, so L2 is odd.
g
To construct Sf
g1 , let Sr,1 be the surface with one boundary component and Euler
L
g
characteristic 2 and let S
s,1 be the surface with one boundary component and Euler
L
g
g
characteristic 2 . By Lemma 23, S
r,1 covers Sr,1 by degree k1 and Ss,1 covers Ss,1 by
g
g
g
g
degree k1 . Let Sf
g = Sr,1 ∪γ˜ Ss,1 be the surface obtained by identifying Sr,1 and Ss,1
where k1 =
L
χ(Sg1 ) .
1
1
along their boundary curves. Then γe1 is a separating curve that projects to γ1 and
f
f
(Sf
g1 , γe1 ) covers (Sg1 , γ1 ) by degree k1 . To see that (Sg , γ) covers (Sg1 , γe1 ), cut Sg1
g
g
along a non-separating curve in S
r,1 and a non-separating curve in Ss,1 and double the
resulting surface with four boundary components. Re-gluing the boundary components
in pairs yields (Sg , γ), as illustrated in Figure 9.
χ(S )
2L
If g2 is odd, (Sg , γ) covers (Sg1 , γ1 ) by degree χ(Sgg ) = χ(S
= 2n d, where n ≥ 1 and
g1 )
1
d is an odd integer. The covering is given via intermediate covers:
2
2
2
d
f
(Sg , γ) −→ . . . −→ (Sc
g1 , γb1 ) −→ (Sg1 , γe1 ) −→ (Sg1 , γ1 ).
23
Figure 9. An example of a multicurve that projects to a non-separating
curve and a separating curve under different covering maps.
d
The first intermediate cover, (Sf
g1 , γe1 ) −→ (Sg1 , γ1 ), is constructed similarly to above.
d·χ(Sg1 )
g
That is, since χ(Sg1 ) is congruent to 2 modulo 4,
is odd. If d > 1, let S
r,1 be the
2
d·χ(Sg1 )
g
surface with one boundary component and Euler characteristic
and let S
s,1 be
2
d·χ(S
)
g1
. By Lemma
the surface with one boundary component and Euler characteristic
2
g
g
f
g
g
23, S
r,1 covers Sr,1 by degree d and Ss,1 covers Ss,1 by degree d. Let Sg1 = Sr,1 ∪γ˜2 Ss,1
g
g
be the surface obtained by identifying S
r,1 and Ss,1 along their boundary curves. Then
f
(Sg1 , γe1 ) covers (Sg1 , γ1 ) by degree d.
2
f
The second intermediate cover, (Sc
g1 , γb1 ) −→ (Sg1 , γe1 ), is the degree two cover of
f
f
g
(Sg1 , γe1 ) obtained by cutting Sg1 along a non-separating curve in S
r,1 and a nong
separating curve in Ss,1 , doubling the resulting surface with four boundary components, and re-gluing along the boundary curves as illustrated in Figure 9. Then, γb1 is
a multicurve that bounds two subsurfaces in Sc
g1 , each with two boundary components
and the same Euler characteristic.
n−1 :
c
If n = 1, then (Sg , γ) = (Sc
g1 , γb1 ). Otherwise, (Sg , γ) covers (Sg1 , γb1 ) by degree 2
cut Sc
b once.
g1 along a non-separating curve that intersects each of the components of γ
Double the resulting surface with two boundary components and re-glue the boundary
components in pairs to form a double cover of (Sc
g1 , γb1 ). Repeat this process n − 1
2
2
times to obtain the cover (Sg , γ) −→ . . . −→ (Sc
g , γb1 ).
1
Suppose now that t(γ1 ) = t(γ2 ) 6= 1 so that Sg1 = Sr1 ,1 ∪γ1 Ss1 ,1 , Sg2 = Sr2 ,1 ∪γ2 Ss2 ,1 ,
and
χ(Sr1 ,1 )
χ(Ss1 ,1 )
=
χ(Sr2 ,1 )
χ(Ss2 ,1 ) .
Construct (Sg , γ) via an intermediate cover
2
k
i
(Sg , γ) −→ (Sg0 , γ0 ) −→
(Sgi , γi ).
Let LR = −`cm(|χ(Sr1 ,1 )|, |χ(Sr2 ,1 )|) and LS = −`cm(|χ(Ss1 ,1 )|, |χ(Ss2 ,1 )|); then LR
and LS are both odd integers. Let SR,1 be the surface with one boundary component
and Euler characteristic LR and let SS,1 be the surface with one boundary component
and Euler characteristic LS . By Lemma 23, SR,1 covers Sri ,1 with degree χ(SLrR,1 ) and
i
24
χ(S
)
χ(S
)
SS,1 covers Ssi ,1 with degree χ(SLsS ,1 ) . Since χ(Ssr1 ,1
= χ(Ssr2 ,1
, χ(SLrR,1 ) = χ(SLsS ,1 ) . Let
1 ,1 )
2 ,1 )
i
i
i
Sg0 = SR,1 ∪γ0 SS,1 , be the surface obtained by identifying SR,1 and SS,1 along their
boundary curves. Then, (Sg0 , γ0 ) covers (Sgi , γi ) by degree ki .
Let (Sg , γ) be the 2-fold cover of (Sg0 , γ0 ) obtained by cutting Sg0 along a nonseparating curve in SR,1 and a non-separating curve in SS,1 and doubling the resulting
surface with four boundary components. Re-gluing the boundary components in pairs
yields (Sg , γ). Then (Sg , γ) covers (Sgi , γi ) and γ is a multicurve with two components,
each of which covers γi by degree ki .
4.2. Abstract commensurability classification. In this section we give the complete abstract commensurability classification within CS . We will use the following
notation in the proof of the theorem: if Sg and Sh are closed surfaces and γg ⊂ Sg and
γh ⊂ Sh are (the images of) multicurves on Sg and Sh , we write (Sg , γg ) ∼
= (Sh , γh ) if
there exists a homeomorphism Sg → Sh mapping γg bijectively to γh .
Theorem 27. If G1 , G2 ∈ CS , then G1 and G2 are abstractly commensurable if and
only if G1 and G2 may be expressed as G1 ∼
= π1 (Sg1 )∗ha1 i π1 (Sg10 ) and G2 ∼
= π1 (Sg2 )∗ha2 i
0
π1 (Sg20 ), given by the monomorphisms ai 7→ [γi ] ∈ π1 (Sgi ) and ai 7→ [γi ] ∈ π1 (Sgi0 ), and
the following conditions hold.
(a)
χ(Sg1 )
χ(Sg2 )
=
,
χ(Sg10 )
χ(Sg20 )
(b) t(γ1 ) = t(γ2 ),
(c) t(γ10 ) = t(γ20 ).
Proof. Let G1 , G2 ∈ CS so that G1 ∼
= π1 (X1 ) and G2 ∼
= π1 (X2 ) with X1 , X2 ∈ XS ,
and suppose G1 and G2 are abstractly commensurable. By Proposition 21, there exist
e1 → X1
finite-index normal subgroups N1 / G1 and N2 / G2 so that N1 ∼
= N2 . Let p1 : X
e2 → X2 be the regular covers corresponding to the subgroups N1 and N2 ,
and p2 : X
e1 ∼
e2 .
respectively. By Theorem 19, X
=X
Let X1 = Sg1 ∪a1 Sg10 where a1 : S 1 → X1 is the image of the essential simple closed
curves γ1 : S 1 → Sg1 and γ10 : S 1 → Sg10 under identification. Then G1 ∼
= π1 (Sg1 ) ∗ha1 i
π1 (Sg10 ) as in the statement of the theorem. Let
S = p−1
1 (Sg1 , γ1 ) =
G
i∈I
(Si , ρi )
and
0
S 0 = p−1
1 (Sg10 , γ1 ) =
G
(Si0 , ρi0 ),
i0 ∈I 0
disjoint unions of closed hyperbolic surfaces Si and Si0 and multicurves ρi and ρi0 . Since
e1 → X1 is regular, there exist pairs of closed surfaces and multicurves
the cover p1 : X
(Sg , γg ) and (Sg0 , γg0 ) so that (Sg , γ) ∼
= (Si , ρi ) and (Sg0 , γ 0 ) ∼
= (Si0 , ρi0 ) for all i ∈ I and
0
0
i ∈I.
Realize X2 as a union of surfaces and G2 as an amalgamated free product as follows.
e1 ∼
e2 , the space X
e2 also consists of the surfaces S and S 0 . Let γ0 : S 1 → X
e1
Since X
=X
be an essential simple closed curve such that p1 (γ0 ) = a1 . Let S0 ∈ S and S00 ∈ S 0 be
such that γ0 ⊂ S0 ∩ S00 . Let Sg2 = p2 (S0 ) and Sg20 = p2 (S00 ), and a2 = p2 (γ0 ) so that
25
X2 = Sg2 ∪a2 Sg20 . Let γ2 : S 1 → Sg2 and γ20 : S 1 → Sg20 map to the inclusion of a2 in
Sg2 and Sg20 . Then G2 ∼
= π1 (Sg2 ) ∗ha2 i π1 (Sg20 ) as in the statement of the theorem.
e2 → X2 , we construct a covering map q2 : X
e2 →
Using the regularity of the cover p2 : X
−1
−1
0
0
0
X2 so that q2 (Sg2 , γ2 ) = S and q2 (Sg20 , γ2 ) = S . We define q2 on S and S and prove
it is well-defined on the overlap. Let fi : (Si , ρi ) → (S0 , ρ0 ) and fi0 : (Si0 , ρi0 ) → (S00 , ρ00 )
be homeomorphisms for all i ∈ I and i0 ∈ I 0 . Let q2 = p2 ◦ fi on S and let q2 = p2 ◦ fi0
on S 0 . The set p−1
2 (a2 ) is a collection of disjoint simple closed curves, each of which
e2 → X2 is a regular cover. Thus, q2 is
projects to a2 by the same degree since p2 : X
e2 .
well defined on the intersection of S and S 0 in X
−1
−1
−1
0
0
Since p−1
1 (Sg1 , γ1 ) = q2 (Sg2 , γ2 ) and p1 (Sg10 , γ1 ) = q2 (Sg20 , γ2 ), claims (b) and (c)
follow from Proposition 26.
e1 is a k1 degree cover of X1 and X
e2 is a k2 degree
To prove claim (a), suppose that X
0
cover of X2 , Sg is an ni degree cover of Sgi , and Sg0 is an ni degree cover of Sgi0 for
ei consists of m copies of Sg and m0 copies of Sg0 . Then,
i = 1, 2. Suppose X
ki = ni m = n0i m0
so that
k1
n1
n0
=
= 01 .
k2
n2
n2
(6)
Also, by Lemma 22,
(7)
χ(Sg ) = n1 χ(Sg1 ) = n2 χ(Sg2 ),
χ(Sg0 ) = n01 χ(Sg10 ) = n02 χ(Sg20 ).
Combining (6) and (7) we have
χ(Sg10 )
χ(Sg1 )
.
=
χ(Sg2 )
χ(Sg20 )
For the converse, we construct a common cover of X1 and X2 when the conditions of the
theorem hold. Let (Sg , γ) be the cover of (Sg1 , γ1 ) and (Sg2 , γ2 ) given by Proposition
26 and let (Sg0 , γ 0 ) be the cover of (Sg10 , γ10 ) and (Sg20 , γ20 ) given by Proposition 26. Let
L = −`cm(|χ(Sg1 )|, |χ(Sg2 )|) and let L0 = −`cm(|χ(Sg10 )|, |χ(Sg20 )|). By construction,
2L
(Sg , γ) covers (Sgi , γi ) by degree χ(S
and each component of γ covers γi by degree
g )
L
χ(Sgi ) .
i
Likewise, (Sg0 , γ 0 ) covers (Sgi0 , γi ) by degree
covers γi0 by degree
L
χ(Sg0 ) .
2L
χ(Sg0 )
and each component of γ 0
i
Let X be the space obtained by identifying Sg and Sg0 along
i
the two components of γ and the two components of γ 0 . By condition (a),
L
L0
=
= ni .
χ(Sgi )
χ(Sgi0 )
Thus, X forms a 2n1 -fold cover of X1 and a 2n2 -fold cover of X2 . Therefore, G1 and
G2 are abstractly commensurable.
26
X1
U
X2
U
X3
U
X4
U
Figure 10. Example: The groups π1 (X1 ), π1 (X2 ), and π1 (X3 ) are abstractly commensurable, but are not abstractly commensurable with π1 (X4 ).
All four groups are quasi-isometric.
4.3. Maximal element in an abstract commensurability class.
Proposition 28. Let G ∼
= π1 (Sg ) ∗hγi π1 (Sg0 ) ∈ CS be given by the monomorphisms
γ 7→ [γg ] ∈ π1 (Sg ), and γ 7→ [γg0 ] ∈ π1 (Sg0 ). Then there exists a maximal element in
the abstract commensurability class of G restricted to CS if and only if γg and γg0 are
separating simple closed curves with Sg = Sr,1 ∪γg Ss,1 , Sg0 = Sr0 ,1 ∪γg0 Ss0 ,1 and r 6= s,
r0 6= s0 .
Proof. We first exhibit a maximal element when the conditions of the proposition hold.
Suppose G ∼
= π1 (Sg ) ∗hγi π1 (Sg0 ) ∈ CS where γg and γg0 are separating simple closed
curves with Sg = Sr,1 ∪γg Ss,1 and Sg0 = Sr0 ,1 ∪γg0 Ss0 ,1 where r 6= s and r0 6= s0 . Let Xg
denote the space obtained by identifying Sg and Sg0 along γg and γg0 . Let C denote the
abstract commensurability class of G restricted to CS . We construct a space X ∈ XS
that Xg covers and prove that if H ∈ C with H ∼
= π1 (Xh ) and Xh ∈ XS , then Xh
covers X as well. Then, π1 (X) is a maximal element in C.
There exist relatively prime p and q and relatively prime p0 and q 0 so that
χ(Sr,1 )
p
=
χ(Ss,1 )
q
and
χ(Sr0 ,1 )
p0
= 0.
χ(Ss0 ,1 )
q
So, χ(Sr,1 ) = −dp, χ(Ss,1 ) = −dq, χ(Sr0 ,1 ) = −d0 p0 , and χ(Ss0 ,1 ) = −d0 q 0 , for some
d, d0 ∈ N. Let k = gcd(d, d0 ). Let Su,1 be the surface with one boundary component
and Euler characteristic − kd p, let Sv,1 be the surface with one boundary component
and Euler characteristic − kd q, let Su0 ,1 be the surface with one boundary component
0
and Euler characteristic − dk p0 , and let Sv0 ,1 be the surface with one boundary com0
ponent and Euler characteristic − dk q 0 . Let S = Su,1 ∪γ Sv,1 be the surface obtained
by identifying Su,1 and Sv,1 along their boundary curves and let S 0 = Su0 ,1 ∪γ 0 Sv0 ,1 be
the surface obtained by identifying Su0 ,1 and Sv0 ,1 along their boundary curves. Then,
27
(Sg , γg ) covers (S, γ) by degree k and (Sg0 , γg0 ) covers (S 0 , γ 0 ) by degree k; so, Xg covers
X by degree k.
To see that π1 (X) is a maximal element C, let H ∼
= π1 (Sh ) ∗hρi π1 (Sh0 ) ∈ C be given by
the monomorphisms ρ 7→ [γh ] ∈ π1 (Sh ) and ρ 7→ [γh0 ] ∈ π1 (Sh0 ). By Theorem 27 (b-i)
and (b’-i’), γh and γh0 are separating simple closed curves so that Sh = Sm,1 ∪γh Sn,1
and Sh0 = Sm0 ,1 ∪γh0 Sn0 ,1 , where
χ(Sm,1 )
p
=
χ(Sn,1 )
q
and
χ(Sm0 ,1 )
p0
= 0.
χ(Sn0 ,1 )
q
Then χ(Sm,1 ) = −f p, χ(Sn,1 ) = −f q, χ(Sm0 ,1 ) = −f 0 p0 , and χ(Sn0 ,1 ) = −f 0 q 0 for some
χ(S )
χ(Sh )
−d(p+q)
−f (p+q)
f
d
, hence −d
f, f 0 ∈ N. By Theorem 27 (a), χ(S g0 ) = χ(S
0 (p0 +q 0 ) = −f 0 (p0 +q 0 ) , so d0 = f 0 .
0)
g
h
0
0
Let ` = gcd(f, f 0 ). By a fact of basic number theory, kd = f` and kd0 = f`0 . Therefore,
(Sh , γh ) covers (S, γ) by degree ` and (Sh0 , γh0 ) covers (S 0 , γ 0 ) by degree `; thus, Xh
covers X by degree ` as desired.
If G ∈ CS does not satisfy the conditions of the proposition, then there are two groups,
H1 and H2 , in the abstract commensurability class of G in CS , where H1 ∼
= π1 (Sh1 )∗hγi
π1 (Sh01 ), H2 ∼
= π1 (Sh2 ) ∗hρi π1 (Sh02 ) and, up to relabeling, γ 7→ [γh1 ] ∈ π1 (Sh1 ), ρ 7→
[γh2 ] ∈ π1 (Sh2 ), where γh1 is an essential non-separating simple closed curve and γh2 is
a separating simple closed curve. Thus, (Sh1 , γh1 ) and (Sh2 , γh2 ) cannot cover the same
pair (S, γ), so there is no maximal element in the abstract commensurability class of
G in CS .
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