Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Classification of Pairs of Fuchsian Groups
S. Allen Broughton
Rose-Hulman Institute of Technology
UNED Geometry Seminar - March 2, 2009
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Outline
1
2
3
4
Introduction/Motivation
Motivation 1 - extension of actions
Motivation 2 - divisible tilings
Motivation 3 - stratification of moduli spaces
Fuchsian Groups - Background
Fuchsian groups
Fuchsian group pairs
Classification Program
Overview
Steps of classification
Constrained and tight pairs
Primitive pairs
Results/References
Current results
Future work
References
Results/References
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Pairs of Fuchsian Groups, Why?
consider finite index pairs Γ ⊆ ∆ of finite area Fuchsian
groups, FG-pairs for short
goal: classify FG-pairs
first consider three motivations
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Motivation 1 - extension of actions
Motivation 1 - extending actions -1
Suppose S is a closed surface with group of
automorphisms H
When is H the full automorphism group?
suppose H admits an overgroup G ⊇ H of automorphisms
there are Fuchsian groups Π ≤ Γ ≤ ∆ such that
Π is torsion free and S w H/Π
Π E Γ and Π E ∆
Γ and ∆ uniformize the action of H and G respectively
namely H and G act on S w H/Π via isomorphisms
H ' Γ/Π and G ' ∆/Π
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Motivation 1 - extension of actions
Motivation 1 - extending actions - 2
uniformization of H-action determined by η : Γ → H
select ∆ from the classification (if it exists, moduli problem)
H-action on S extends to G ' ∆/Π if there is an extension
η:∆→G
whether η extends can be determined by knowing the
inclusion map Γ → ∆ in algebraic form
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Motivation 1 - extension of actions
Motivation 1 - finitely maximal - 1
Γ is finitely maximal if there is no overgroup of finite index
action can extend only in case Γ is not finitely maximal
Greenberg and later Singerman worked out a finite list of
Fuchsian groups that are never finitely maximal.
action extension is now just an algebraic problem, no
moduli constraints
this case by worked out by Bujalance, Conder and Cirre
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Motivation 2 - divisible tilings
Motivation 2 - divisible tilings 1
suppose P is a convex hyperbolic polygon whose angles
have the form π/n for n ∈ {2, 3, 4, . . . , ∞}
called a kaleidoscopic polygon, see slide after next
P generates a tiling by repeated reflection in sides
reflections in sides of polygon generate a crystallographic
group
define Γ to be the subgroup of orientation preserving
transformations
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Motivation 2 - divisible tilings
Motivation 2 - divisible tilings 2
suppose Q ⊂ P is a kaleidoscopic polygon whose tiling
refines first tiling, namely P is tiled by repeated reflections
of Q
let ∆ be the Fuchsian group determined by Q
then Γ ⊂ ∆ is a FG-pair where the index is the number of
Q-polygons tiling a P-polygon
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Motivation 2 - divisible tilings
Motivation 2 - divisible tilings 3
P = large polygon, angles = (π/5, π/5, π/5, π/5)
Q = small polygon, angles = (π/2, π/4, π/5)
Results/References
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Motivation 2 - divisible tilings
Motivation 2 - divisible tilings 4
P = large polygon, angles = (π/3, π/4, π/5, π/20)
Q = small polygon, angles = (π/2, π/3, π/20)
Results/References
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Motivation 2 - divisible tilings
Motivation 2 - divisible tilings 5
in the last frame observe that there are “freely moveable”
vertices
we get an entire family of polygon pairs Q ⊂ P where P
has angles (π/3, π/4d, π/5d, π/20d), Q has angles
(π/2, π/3, π/20d), and d ≥ 1 is an integer parameter
the limiting polygon as d → ∞ has cusps on the boundary
of H
the entire family is determined by the limiting cusp case
the algebraic structure of ∆/Γ is independent of the order
at the free vertices.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Motivation 3 - stratification of moduli spaces
Motivation 3 - stratification of moduli space 1
moduli space Mσ is the space of conformal equivalence
classes of surfaces of genus σ
Mσ may be stratified into a finite disjoint union of locally
closed smooth subvarieties, consisting of surfaces of
similar symmetry type, called equisymmetric strata
all surfaces in a given stratum have automorphism group
isomorphic to a fixed group H (all conjugate in the mapping
class group)
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Motivation 3 - stratification of moduli spaces
Motivation 3 - stratification of moduli space 2
if Σ is a stratum its closure is a union of strata of lower
dimension (adjunction) corresponding to overgroups
G⊇H
G ⊇ H determines an FG-pair Γ ⊂ ∆
classification of the FG-pairs helps determine the
equisymmetric structure of Mσ (adjunction relation)
the union of the strata for non trivial groups form the
so-called branch locus of moduli space.
related to current work of Costa and Milagros
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian groups
Fuchsian groups - presentation
Γ has a presentation
generators : {αi , βi , γj , δk , 1 ≤ i ≤ σ, 1 ≤ j ≤ s, 1 ≤ k ≤ p}
p
σ
s
Y
Y
Y
relations :
[αi , βi ]
γj
δk = γ1m1 = · · · = γsms = 1
i=1
j=1
k =1
the signature of Γ is
S(Γ) = (σ : m1 , . . . , ms , ms+1 , . . . , ms+p )
with ms+j = ∞, j = 1, . . . , p
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian groups
Fuchsian groups - invariants
important invariants of a Fuchsian group
the genus of Γ: σ(Γ) = σ is the genus of S = H/Γ
area of a fundamental region: A(Γ) = 2πµ(Γ) where:
µ(Γ) = 2(σ − 1) +
s+p
X
j=1
(1 −
1
).
mj
Teichmüller dimension d(Γ) of Γ: the dimension of the
Teichmüller space of Fuchsian groups with signature S(Γ)
given by
d(Γ) = 3(σ − 1) + s + p.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - index and codimension
for finite index FG-pair Γ ≤ ∆, [∆ : Γ] = µ(Γ)/µ(∆)
for finite index FG-pair Γ ≤ ∆, we call the quantity
d(Γ, ∆) = d(Γ) − d(∆) the Teichmüller codimension of
(Γ, ∆)
in Motivation 1: Singerman’s list is the list of codimension 0
pairs.
in Motivation 2: the tiling of an m-gon by an n-gon
determines a codimension m − n pair
in Motivation 3: If Σ2 lies in the closure of Σ1 then a
codimension dim(Σ1 ) − dim(Σ2 ) pair is determined
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - generating sets
Let Γ ≤ ∆ be as before and suppose that Γ has genus σ, s
elliptic generators, and p parabolic generators and that ∆
has genus τ , t elliptic generators, and q parabolic
generators.
For the remainder of the discussion let Γ and ∆ have
canonical generating sets
G1 = {θ1 , . . . , θ2σ+s+p }
and
G2 = {ζ1 , . . . , ζ2τ +t+q },
respectively.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - induced generators 1
an elliptic or parabolic generator θ ∈ Γ is induced by the
generator ζ ∈ ∆ (θ lies over ζ) if
θ = xζ k x −1
for some x ∈ ∆ and k > 0
the property of lying over is completely captured by the
action of hζi on ∆/Γ or alternatively by the ramification
structure of π : H/Γ → H/∆,
k = | hζi xΓ| in ∆/Γ and equals the local ramification
degree of H/Γ → H/∆ at the fixed point of θ
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - induced generators 2
if k = | hζi xΓ| = o(ζ), then x hζi x −1 ∩ Γ is trivial and there
is no θ.
the condition
o(ζ) = ko(θ)
holds, with o(ζ) = o(θ) = ∞ for the parabolic case
the elliptic or parabolic generators θ ∈ Γ are in 1-1
correspondence to the collection of orbits of elliptic or
parabolic ζ ∈ ∆ that satisfy
| hζi xΓ| < o(ζ)
show example for S(Γ) = (0; 2, 2, 2, 5) and
S(∆) = (0; 2, 4, 5)
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - monodromy 1
the pair Γ ≤ ∆ of index m determines a permutation
representation ρ : ∆ → Σm where m is the index of Γ in ∆
the representation may be captured geometrically from the
monodromy of the branched covering S → T , S = H/Γ,
T = H/∆
M(∆, Γ) = ρ(∆) is called the monodromy group of the pair
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - monodromy 2
a generating set for ∆ determines a sequence
P = {ρ(αi ), ρ(βi ), ρ(γj ), ρ(δk ), 1 ≤ i ≤ τ, 1 ≤ j ≤ t, 1 ≤ k ≤ q}
of elements of Σm satisfying certain properties
the sequence is called a monodromy vector
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - monodromy 3
Write P = (π1 , π2 , . . . , π2τ +t+q ) for πi ∈ Σm ,
the monodromy vector P satisfies
hπ1 , π2 , . . . , π2τ +t+q i is a transitive subgroup of Σm
τ
Q
[πi , πτ +i ]
i=1
t+q
Q
π2τ +j = 1
j=1
the signatures S(Γ), S(∆) determine the cycle types of the
permutations corresponding to elliptic and parabolic
generators
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - monodromy 4
let πi be the monodromy image of an elliptic or parabolic
generator ζ
let pi = (pi,1 , . . . , pi,r ) be the partition of n determined by πi
for each pi,j there is a period mi,j of Γ such that
o(ζ) = pi,j mi,j
review example for S(Γ) = (0; 2, 2, 2, 5) and
S(∆) = (0; 2, 4, 5)
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Fuchsian group pairs
Fuchsian group pairs - monodromy 5
Riemann Existence Theorem
P a monodromy vector compatible with Γ < ∆
ρ : ζi → πi determines a homomorphism ρ : ∆ → Σm
Define
Γ = {g ∈ ∆ : ρ(g) · 1 = 1}
Γ has the right signature - compatible with P
Results/References
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - word maps
canonical generating sets of Γ and ∆ are given as before:
G1 = {θ1 , . . . θ2σ+s+p } and G = {ζ1 , . . . ζ2τ +t+q }
respectively.
The word map of the pair Γ ≤ ∆ is a set of words
{w1 , . . . , w2σ+s+p } in the generators in G2 such that
θi = wi (ζ1 . . . , ζ2τ +t+q ), i = 1, . . . , 2σ + s + p
If both groups have genus zero there is an easily
implemented algorithm to calculate the word map.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - example 1
Suppose we have these signatures
S1 = (0; 2, 2, 2, 5), S2 = (0; 2, 4, 5)
Show there is a pair Γ < ∆ with
S(Γ) = S1 , S(∆) = S2
find compatible monodromy vector in Σ6
π1 = (1, 3)(4, 6), π2 = (1, 2)(3, 5, 4, 6), π3 = (1, 2, 3, 4, 5)
M(∆, Γ) = A6
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Fuchsian group pairs
Fuchsian group pairs - example 2
define as before ρ : ∆ → Σ6 by ρ : ζi → πi , i = 1 . . . 3
Γ is the stabilizer of a point for the permutation action of ∆
on {1, . . . , 6}
From the algorithm, a generating set for Γ is
θ1
θ2
θ3
θ4
= (ζ1 ζ2 )ζ1 (ζ1 ζ2 )−1
= ζ2 ζ1 ζ2−1
= ζ22
= (ζ2−1 ζ1−1 ζ2−1 ζ1 ζ3 ζ1 )ζ3 (ζ2−1 ζ1−1 ζ2−1 ζ1 ζ3 ζ1 )−1
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Overview
Overview
classify by codimension
constrained and tight pairs
finiteness of classification in each codimension
numerical projections
primitive pairs, M(∆, Γ) is a primitive permutation group
big and small primitive monodromy groups
towers of Fuchsian groups
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Steps of classification
Steps of classification - 1
determine all signature pairs for a fixed codimension
can be done by computer search resulting in
finitely many exceptional cases (constrained pairs)
finitely many families (tight pairs)
Results/References
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Steps of classification
Steps of classification - 2
for each candidate signature pair, compute all the
compatible monodromy vectors up to conjugacy
equivalence
use computer calculation and classification of primitive
permutation groups (use Magma or GAP)
using the Riemann existence theorem, each monodromy
vector determines an FG-pair
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Steps of classification
Steps of classification - 3
for each monodromy vector a branched cover
π : H/Γ → H/∆ may be constructed
the word map Γ ,→ ∆, written in terms of a canonical
generating set, may be constructed from the branched
cover
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Steps of classification
Steps of classification - 4
the general case defines a tower Γ = Γ1 < · · · < Γe = ∆
such that each inclusion Γi < Γi+1 is a primitive pair,
already classified
Example: T7,7,7 < T3,3,7 < T2,3,7
possible towers to examine obtained by factorization of
numerical projections
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Constrained and tight pairs
Constrained and tight pairs - 1
Definition
Let ρ : ∆ → Σn as previously defined.
A pair Γ < ∆ is called constrained if ∆ has no parabolic
generators and o(ζ) = o(ρ(ζ)) for each elliptic generator.
A pair Γ < ∆ is called tight if ∆ has at least one parabolic
generator and o(ζ) = o(ρ(ζ)) for each elliptic generator.
Remark
The definition depends only on the cycles types and not the
permutations themselves. Hence, the definition depends only
on the signature pair.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Constrained and tight pairs
Constrained and tight pairs - 2
Proposition
Let Γ < ∆ be a tight pair where ∆ has q parabolic elements.
Then there is a q parameter family Γ(`1 , . . . , `q ) < ∆(`1 , . . . , `q )
such that each member of the family has
the same codimension d(Γ, ∆)
the same index [∆ : Γ]
the same monodromy M(∆, Γ)
Remark
Every Fuchsian group pair is constrained or belongs to a
unique family as above. The tight pair defining the family is
called the parent tight pair.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Constrained and tight pairs
Constrained and tight pairs - 3
Example
The triangle group family T (2, d, 2d) < T (2, 3, 2d) comes from
the tight pair T (2, ∞, ∞) < T (2, 3, ∞). The monodromy vector
is ((1, 2), (1, 2, 3), (1, 3)).
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Constrained and tight pairs
Numerical projections - 1
The ramification information of π : H/Γ → H/∆, does not
need the monodromy vector to be described just the cycle
types of the monodromy images of the elliptic an parabolic
generators.
Let P = (p1 , . . . , pt+q ) be the vector of partitions defined by
the monodromy images of the elliptic and parabolic
generators.
P : S(Γ) → S(∆) is called a numerical projection.
called realizable if P is derived from a monodromy vector
of a pair Γ < ∆.
an example is (12 · 22 , 2 · 4, 1 · 5) : (0; 2, 2, 2, 5) → (0; 2, 4, 5)
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Primitive pairs
Primitive pairs
If there are no intermediate groups Γ < Γ0 < ∆ then the
monodromy group M(∆, Γ) is a primitive group.
Primitive groups are known for degree up to 1000 or so.
But transitive groups are only catalogued to degree 30 or
so.
Therefore, only primitive groups are determined. The
non-primitive pairs can be determines as a tower of
primitive pairs.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Primitive pairs
Primitive pairs - towers 1
A tower of primitive pairs is a sequence
Γ = Γ1 < · · · < Γe = ∆
such that each inclusion Γi < Γi+1 is a primitive pair.
If a tower exists then there is a corresponding factorization
of the numerical projection.
Towers of groups may be found as follows.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Primitive pairs
Primitive pairs - towers 2
Find all possible factorizations of the numerical projection.
For a given factorization check that all factors are
realizable.
Construct the word maps corresponding to the monodromy
vectors.
Compose the word maps to get word map for Γ < ∆
Use the Todd-Coxeter algorithm to compute the
monodromy of Γ < ∆ if needed.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Primitive pairs
Primitive pairs - big and small monodromy groups
The primitive groups of a fixed degree are either “small” or
“big” i.e., alternating and symmetric.
For big monodromy groups search methods for
monodromy vectors fail because it takes too long to
complete.
At present only results for small monodromy groups seems
possible.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Current results
Finiteness of the classification
Theorem
For each codimension there is a finite number of constrained
pairs and a finite number of tight pairs. For each signature pair
there is a most a finite number of group pairs up to monodromy.
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
Current results
Results by codimension
Codimension 0 - Singerman’s list - inclusions have been
calculated - Bujalance, Conder and Cirre
Codimension 1 - all cases coming from polygonal pairs
determined by previous efforts of author and students,
inclusions not yet calculated
for codimension 0,1,2,3 all signature pairs calculated
codimension codimension 0,1,2,3 algorithm for computing
realizable projections with small groups are essentially
done.
algorithm for computing word map when both groups have
genus zero has been completed
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Future work
future work
complete calculations
compute large monodromy groups
find general algorithm for word map
Results/References
Introduction/Motivation
Fuchsian Groups - Background
Classification Program
Results/References
References
References
On extendability of group actions on compact Riemann
surfaces, Bujalance, Conder, and Cirre, Trans. Amer.
Math. Soc. 355 (2003), 1537-1557.
Divisible Tilings in the Hyperbolic Plane (with Dawn M.
Haney, Lori T. McKeough, Brandy M. Smith), New York
Journal of Mathematics 6 (2000), 237-283.
http://www.rose-hulman.edu/vbrought/
D. Singerman. Finitely Maximal Fuchsian Groups, J.
London Math. Society(2) 6, (1972),17-32