## Enumeration of the
Equisymmetric Strata of the
Moduli Space of Surfaces of
Low Genus

### - preliminary report-

at the

### ** Univerity of Santa Barbara, Santa Barbara, CA (2005 Spring Western Section
Meeting) Meeting # 1007 **

### Abstract

Two surfaces are called equisymmetric, or are said to have the same symmetry
type, if the two surfaces’ conformal
automorphism groups determine conjugate finite subgroups of the mapping class
group. The subset of the moduli space
corresponding to surfaces equisymmetric with given surface forms a locally closed
subvariety of the moduli space, called
an equisymmetric stratum. In previous work, it was shown that the equisymmetric
strata are irreducible, finite in number,
have easily computed dimensions, and do form a stratification of the moduli space.
The stratification has been used to
derive information about the cohomology of the mapping class group. Recent advances
in computer calculation with
finite groups allow for the possibility of explicitly enumerating the equisymmetric
strata for moduli spaces of low genus.
In this preliminary report we give some initial findings on this enumeration
problem.
### Lecture and other materials

### Note

One of my references on which the talk is based has an omission. The paper
is

- Classifying Finite Group Actions on
Surfaces of Low Genus, Journal of Pure and Applied Algebra 69 (1990).
- The missing group is in genus 3. The group is called SmallGroup(48,33) in
the GAP/MAGMA system. The signature of the corresponding Fuchsian group is
(2,3,12) and the triple of elements defining the action is (g1g3, g2, g1g2g3).
- The PC group definition is:
- GrpPC : G of order 48 = 2^4 * 3,

Generators: g1, g2, g3, g4, g5

PC-Relations:

g1^2 = g5,

g2^3 =1,

g3^2 = g5,

g4^2 = g5,

g5^2 = 1,

g3^g2 = g4,

g4^g2 = g3 g4,

g4^g3 = g4 g5

### Links

Email: allen.broughton@rose-hulman.edu
Webpage: http://www.rose-hulman.edu/~brought/
This page last updated on 18 Apr 05.