Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
Elementary Abelian Group Actions on
Surfaces and the Geometry of Moduli Space
S. Allen Broughton - Rose-Hulman Institute of Technology
Aaron Wootton - University of Portland
IU Geometry Seminar Nov 8, 2007
References
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Overview
parts of the talk
There are three main parts of the talk
group actions and the mapping class group - background,
mostly topology, some geometry
Moduli spaces, Teichmüller space and the mapping class
group - background, some geometry in this part
classification of elementary abelian actions - new results,
all algebra
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Motivation
motivation - automorphisms of surfaces
The study of automorphisms of surfaces is over 100 years
old but still fertile territory.
Steady advances from original ad-hoc methods to
Fuchsian group methods, moduli spaces and Teichmüller
spaces.
Recent advances in computer algebra systems (GAP or
Magma) make discovery/classification of doable and of
interest. Many old questions are now accessible.
For example T. Breuer has a classification up to homology
equivalence for genus up to 48.
Elementary abelian actions are a tractable first step
towards classification.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Motivation
motivation - moduli space
The moduli space is easily defined but its geometric
structure -specifically it singularity structure - is unknown
except for low genus.
The classification of groups of automorphisms of surface
and more specifically the classification of conjugacy
classes of finite subgroups of the mapping class is directly
relevant to this question.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Motivation
motivation - cohomology
The finite elementary abelian subgroups of infinite discrete
groups play a strong role in the cohomology of infinite
discrete groups over finite fields.
So the analysis of finite elementary abelian actions of
surfaces play a role in the study of cohomology of mapping
class groups.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Group actions on surfaces
topological action and equivalence
Definition
Let G be a finite group.
The group G acts (orientation preserving) on a closed,
orientable surface S of genus σ ≥ 2 if there is an injection
: G ,→ Homeo+ (S)
into the group of orientation preserving homeomorphisms.
Two actions 1 , 2 are topologically equivalent if there is a
homeomorphism h of S and an automorphism ω of G such
that
2 (ω(g)) = h ◦ 1 (g) ◦ h−1 , g ∈ G.
i.e., 1 (G) and 2 (G) are conjugate in Homeo+ (S).
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Group actions on surfaces
conformal action and equivalence
Definition
Let G be a finite group and S a closed, orientable surface S
with a complex analytic structure. Then
the group G acts conformally on S if there is an injection
: G ,→ Aut(S)
into the group of conformal automorphisms of S.
Two actions 1 , 2 are conformally equivalent if there is a
conformal automorphism h of S and an automorphism ω of
G such that
2 (ω(g)) = h ◦ 1 (g) ◦ h−1 .
i.e., 1 (G) and 2 (G) are conjugate in Aut(S).
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Group actions on surfaces
remarks and examples
In the definition of topological actions we can replace
Homeo+ (S) by Diff+ (S).
In the definition of conformal actions we can replace
“complex analytic structure” by “metric of constant
curvature” and “automorphism” by “isometry”.
Suppose that S has genus σ ≥ 2 and that has a conformal
structure. Then any group of automorphisms of S is finite.
(Old theorem, consider action on Weierstrass points).
If finite G acts on a surface S, then S may be given a
conformal structure so that G is a group of automorphism.
Nielsen Realization problem. Proven by S. Kerckhoff.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Mapping class group (MCG)
definition of MCG
Definition
Let S be an orientable closed surface of genus σ. The mapping
class group Mσ is the group of homotopy classes of orientation
preserving homeomorphisms of S. Equivalently let Homeo+
0 (S)
be the identity component subgroup of Homeo+ (S), then
Mσ = Homeo+ (S)/Homeo+
0 (S)
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
Mapping class group (MCG)
remarks and examples
Lots is known on mapping class groups - see Birman’s
book.
generators and relations
cohomology
various subgroups
actions on various spaces
M1 = PSL(2, Z)
Mσ for σ ≥ 2 is harder to describe explicitly.
References
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Mapping class group (MCG)
finite subgroups of MCG
Proposition
Let finite G act on a surface S of genus σ ≥ 2. Then the
map
: G ,→ Homeo+ (S) → Mσ
is injective.
Topological equivalence classes of finite group actions are
in 1-1 correspondence to conjugacy classes of finite
subgroups of the mapping class group.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
Mapping class group (MCG)
finite subgroups of MCG
Genus 2 and 3 have been completely done by hand.
There are 20 classes of so for genus 2.
There are 53 classes of so for genus 3.
References
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Mapping class group (MCG)
homology equivalence -1
Consider the map
ρ : Mσ → GL2σ (C)
coming from the homology representation.
The restriction to a finite subgroup
ρ : G → Mσ → GL2σ (C)
is still injective.
There are non-conjugate subgroups of Mσ with conjugate
images in GL2σ (C).
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Mapping class group (MCG)
homology equivalence - 2
Two group actions 1 , 2 are homology equivalent if there is
an automorphism ω of G and a matrix M in GL2σ (C) such
that
ρ(2 (ω(g))) = Mρ(1 (g))M −1 , g ∈ G.
Breuer uses this weaker equivalence in his classification.
Example: There are three non-conjugate subgroups of M14
isomorphic to PSL(2, 13), with conjugate images in
GL26 (C). The homology representations are equivalent
over C.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Construction of actions
uniformization of actions
Denote the hyperbolic plane (upper half complex plane) by
H. Its automorphism group Aut(H) equals PSL2 (R) acting
by fractional linear transformations.
A Fuchsian group is a discrete subgroup of
Aut(H) = PSL2 (R).
Proposition
Let G act conformally on S. Then, there are Fuchsian groups
Π ⊆ Γ such that
Π ' π1 (S), S ' H/Π, and
G ' Γ/Π and G acts on S via the natural action of Γ/Π on
S ' H/Π.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Construction of actions
Fuchsian group generators
S → S/G = T derived from H/Π → H/Γ is a branched
covering, branched over t points with local branching
orders m1 , . . . , mt .
Let τ = genus of T and call S = (τ ; m1 , . . . , mt ) the
branching data of the G-action or the signature of Γ.
The uniformizing Fuchsian group Γ has a generating set of
the following form.
generators : {αi , βi , γj , 1 ≤ i ≤ τ, 1 ≤ j ≤ t}
τ
t
Y
Y
relations :
[αi , βi ]
γj = γ1m1 = · · · = γtmt = 1
i=1
j=1
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Construction of actions
generating vectors - 1
An action gives a sequence, called a surface-kernel
epimorphism,
η
Π ,→ Γ G
Set
ai = η(αi ), bi = η(βi ), cj = η(γj )
and call the (2τ + t)-tuple (a1 , . . . aτ , b1 , . . . bτ , c1 , . . . ct ) a
“(G, S)-vector”.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
Construction of actions
generating vectors - 2
The constructed vector satisfies
G = ha1 , . . . aτ , b1 , . . . bτ , c1 , . . . ct i
t
τ
Y
Y
[ai , bi ]
cj = 1,
i=1
j=1
o(cj ) = mj .
Any vector satisfying the above is called a generating
(G, S)-vector for G acting on S.
We also have the Riemann Hurwitz equation which is a
constraint between σ and (G, S).
t
X 1
(2σ − 2)
= (2τ − 2 + t) −
|G|
mj
j=1
References
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Construction of actions
generating vectors - 3
Let S = (τ ; m1 , . . . , mt ) and let X ◦ (G, S) be the set of
generating (G, S)-vectors of G.
X ◦ (G, S) is in 1-1 correspondence to Epi ◦ (Γ, G) which is
η
the set of epimorphisms Π ,→ Γ G with torsion free
kernel.
Aut(G) × Aut(Γ) acts on Epi ◦ (Γ, G) and hence X ◦ (G, S) by
η −→ ω ◦ η ◦ ξ −1
where (ω, ξ) ∈ Aut(G) × Aut(Γ).
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Construction of actions
reduction to algebra theorem
Proposition
The topological equivalence classes of S-actions of G on a
surface S are in 1-1 correspondence with the Aut(G) × Aut(Γ)
orbits on X ◦ (G, S). The genus of S must satisfy the
Riemann-Hurwitz equation.
Remark
The calculation of Aut(G)-action is pretty straight forward
The calculation of Aut(Γ)-action can be ghastly, but it is not
too bad in the abelian case. See Broughton-Wootton paper
for formulas.
The formulas for the Aut(Γ)-action can be derived from
looking at Dehn twists and spin maps on T = S/G that
preserve branching order.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
Moduli space
moduli and rigidity
Moduli Spaces
Surfaces are the only examples of non-rigid compact
hyperbolic manifolds. (Mostow Rigidity Theorem).
Homeomorphic surfaces are not necessarily isometric.
Use the moduli space to measure variation of isometry
class.
References
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
Moduli space
definition of moduli space
Definition
The moduli space Mσ is the set of conformal equivalence
classes (isometry classes) of closed, orientable surfaces of
genus σ.
References
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Moduli space
structure of moduli space
Proposition
The moduli space Mσ has the structure of a quasiprojective
variety. Namely there are closed sets X , Y , defined by the
vanishing of polynomials, in a very large projective space
PN (C), such that
Mσ = X − Y .
Remark
The moduli space has a decent compactification by adding
surfaces with mild singularities.
The moduli space has a enormous number of singularities
and singular subvarieties. Gaining some understanding of
this structure is the point of this study.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Teichmüller space
definition of Teichmüller space
Here is a reasonable definition of Teichmüller space Tσ .
Let Π ' π1 (S) be the Fuchsian group defined earlier.
Definition
e σ = {r : Π → PSL2 (R) | r (Π) is discrete, r is injective}
T
e σ /PSL2 (R)
Tσ = T
.
Using the canonical generating set for Π we can show that
e σ is a subset in (PSL2 (R))2σ and so Tσ is a manifold of
T
finite dimension.
There is a map Tσ → Mσ given by r → class of H/r (Π).
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Teichmüller space
complex structure of Teichmüller space
Proposition
Tσ is a complex analytic manifold, isomorphic to an open region
in C3σ−3 and homeomorphic to R6σ−6 .
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
MCG action, branch loci, and equisymmetric strata
action and quotient by the MCG
Each element h of the MCG defines an automorphism h∗
of Π which is unique up to an inner automorphism.
e σ and hence Tσ by r → r ◦ h−1
In turn h∗ acts on T
∗
Each automorphism of Π is induced by an element of the
MCG.
Thus Mσ ' Out(Π) acts on Tσ .
The action of Mσ on Tσ is a discontinuous action, and
Tσ /Mσ = Mσ
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
MCG action, branch loci, and equisymmetric strata
branch loci
A subgroup of Mσ fixes a point in Tσ if and only if the
subgroup is finite.
For a finite subgroup the fixed point subset is called the
branch locus. The branch locus is isomorphic to a
Teichmüller space of complex dimension 3τ − 3 + t. We
call 3τ − 3 + t the Teichmüller dimension of G.
The hyperelliptic locus is the fixed point subset (and its
image in Mσ ) corresponding to the conjugacy class of a
hyperelliptic involution. This branch locus has codimension
σ−2
Except along the hyperelliptic locus for genus 2 and 3, Mσ
is singular along the image of any branch locus. The
singularity structure of Mσ is captured by the conjugacy
classes of the mapping class group, though the
correspondence is not 1-1.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
MCG action, branch loci, and equisymmetric strata
branch loci and equisymmetric strata
Pick a surface S and let G be a finite subgroup of the MCG
determined by the automorphism group of S.
Let TG be the corresponding branch locus.
Cut out from TG all TH for subgroups H strictly containing
G resulting in T◦G .
The image M◦G of T◦G in Mσ is the set of all surfaces whose
automorphism group is conjugate to G in the MCG.
Call each such object T◦G an equisymmetric stratum.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
MCG action, branch loci, and equisymmetric strata
decomposition of moduli space
Proposition
Moduli space is a disjoint union
Mσ =
[
M◦G
G
of equisymmetric strata over conjugacy classes of "finitely
maximal" subgroups. The strata are all smooth, irreducible and hence connected - subvarieties of moduli space. A finite
subgroup (actually its covering Fuchsian group Γ) is "finitely
maximal" if and only if it is not contained in any larger finite
group with the same Teichmĺler dimension.
References
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Simplifications in the elementary abelian case
elementary abelian actions - simplifications - 1
Elementary abelian actions
Suppose p is a prime and G is an
elementary abelian group of rank v . G is a
vector space of dimension v over Fp
If S = (τ ; −) then the action is called
unramified.
If S = (0; pt ) (sphere with t branch points of
order p then the action is called purely
ramified.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Simplifications in the elementary abelian case
elementary abelian actions - simplifications - 2
If the action is unramified then the Aut(Γ) action is the
Sp(2τ, Fp ) action coming from the homology
representation of MCG(S/G) = Out(Γ).
If the action is purely ramified then the Aut(Γ) action is the
symmetric group permuting the branch points and hence
permuting the elements cj .
To see the last bullet, observe that
(c1 , . . . , cj , cj+1 , . . . , ct ) → (c1 , . . . , cj cj+1 cj−1 , cj , . . . , ct )
is always a permissable transformation if the orders are the
same. The transformation is obtained by spinning one
branch point about the other. For abelian groups the
conjugates go away and we get a transposition.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Simplifications in the elementary abelian case
elementary abelian actions - simplifications - 3
The following simplifies matters
Proposition
Every elementary abelian action is a direct sum of an
unramified action and a purely ramified action
Apply transformations of the following form to a generating
vector (a1 , . . . aτ , b1 , . . . bτ , c1 , . . . ct )
ai → ai cjk ,
bi 0 → bi 0 cjk0
0
until the modified vector satisfies
G = ha1 , . . . aτ , b1 , . . . bτ i
M
hc1 , . . . ct i
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Classification
unramified
Proposition
Every unramified action of G = Cpv is equivalent to one
constructed from a generating vector of the following type.
Select two integers d, e, 0 ≤ d ≤ e ≤ τ , and set
ai = x, 1 ≤ i ≤ e,
and
bi = x, 1 ≤ i ≤ d,
all other generators are trivial, and x is a generator of Cp , the
cyclic group of order p.
The rank of G will be v = 2d + e and the genus of S is given by
the Riemann-Hurwitz formula.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Classification
purely ramified
This case is much more complex since the “action image"
of Aut(G) × Aut(Γ) on X ◦ (G, S) is much smaller than in the
ramified case, and consequently there are many orbits.
A methodology was developed that allows complete
classification, provided one can do the calculations.
Methods make use of the following techniques.
Use the vector space structure of G.
Analysis of representation theory of Aut(G) × Aut(Γ) on the
ambient vector space in which X ◦ (G, S) resides. This boils
down to an analysis of the representation theory
(sometimes modular) of the subgroups of the symmetric
group.
Repeated use of Möbius inversion, inclusion-exclusion and
other combinatorial counting techniques.
Show table.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Example
hyper-Fermat curves - 1
The Fermat curve z1p + z2p = 1 has a G = Cp × Cp action
by multiplying componentwise by p’th roots of unity. The
projective completion, Fp , of this complex curve is a
smooth, connected surface and G has branching data
(0; p, p, p)
For appropriate coefficients the complex curve defined by
a1 z1p + a2 z2p + a3 z3p = 1
b1 z1p + b2 z2p + b3 z3p = 1
has a smooth, connected projective completion with a
G = Cp × Cp × Cp action. G has branching data
(0; p, p, p, p)
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
Example
hyper-Fermat curves -2
The construction can be repeated for all branching data
(0; pt ) for t > 2.
The group is Cpt−1 and the genus is given by
t−2 t(p − 1) − 2p
σ =1+p
2
Moreover, the complex dimension of this stratum in the
moduli space is t − 3 and all surfaces in the stratum can be
constructed as above.
Introduction
Surface actions and mapping class group
Moduli space
Elementary abelian actions
References
References
J. Birman, Braids, Links and Mapping Class Groups,
Annals of Math. Studies, No. 82, Princeton U. Press
(1974).
T. Breuer. Characters and Automorphism Groups of
Compact Riemann Surfaces, Cambridge University Press
(2001).
S.A. Broughton. The equisymmetric stratification of the
moduli space and the Krull dimension of the mapping class
group, Topology and its Applications, 37 (1990), 101-113.
S. A. Broughton, A. Wootton, Finite Abelian Subgroups of
the Mapping Class Group, preprint available at
http://arxiv.org/abs/math.AT/0611650