Group Actions Seminar: Gebhardt, Egri-Nagy

The next Group Actions Seminar will be held on Tuesday 3 April at the University of
Western Sydney.  Volker Gebhardt will speak at 12 noon and Attila Egri-Nagy at 2:30pm.

**Note that the seminar is being held at the Kingswood campus of UWS,
not at Parramatta as previously advertised.**

If you would like to get the train from Central with me, please let me know.

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Date: Tuesday 3 April

Time: 12 noon

Location: Room 2.39, Building Y, Kingswood Campus, University of Western Sydney

Speaker: Volker Gebhardt (University of Western Sydney)

Title: Finite index subgroups of mapping class groups

Abstract:

The interaction between mapping class groups and finite groups has long been a
topic of interest. The famous Hurwitz bound of 1893 states that a
closed Riemann surface of genus g has an upper bound of 84(g-1) for
the order of its finite subgroups, and Kerckhoff showed that the order
of finite cyclic subgroups is bounded above by 4g+2.

The subject of finite index subgroups of mapping class groups was
brought into focus by Grossman’s discovery that the mapping class
group M_{g,n} of an oriented surface \Sigma_{g,n} of genus g with n
boundary components is residually finite, and thus well-endowed with
subgroups of finite index.  This prompts the "dual’’ question:

What is the minimum index mi(M_{g,n}) of a proper subgroup of finite
index in  M_{g,n}?

Results to date have suggested that, like the maximum finite order
question, the minimum index question should have an answer that is
linear in g. The best previously published bound due to Paris is
mi(M_{g,n})>4g+4 for g \geq 3. This inequality is used by Aramayona
and Souto to prove that, if g \geq 6 and g’ \leq 2g-1, then any
nontrivial
homomorphism M_{g,n} \to M_{g’,n’} is induced by an embedding. It is
also an important ingredient in the proof of Zimmermann that, for g=3
and 4, the minimal nontrivial quotient of M_{g,0} is Sp_{2g}(F_2).

I will report on recent work with Jon Berrick and Luis Paris, in which we showed
an exact exponential bound for mi(M_{g,n}).  Specifically, we proved
that M_{g,n}  contains a unique subgroup of index 2^{g-1}(2^{g}-1) up
to conjugation, a unique subgroup of index 2^{g-1}(2^{g}+1) up to
conjugation, and the other proper subgroups of M_{g,n} are of index
greater than 2^{g-1}(2^{g}+1). In particular, the minimum index for a
proper subgroup of M_{g,n} is 2^{g-1}(2^{g}-1).

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Date: Tuesday 3 April

Time: 2.30pm

Location: Room 2.39, Building Y, Kingswood Campus, University of Western Sydney

Speaker: Attila Egri-Nagy (University of Western Sydney)

Title: Recursive Reduction of Multiplication Tables

Abstract:

Motivated by the problem of enumerating finite transformation
semigroups here we present a recursive algorithm for systematic
reduction of multiplication tables. We can downsize the hopelessly
huge search space even if we do not exploit properties of the
underlying algebra. By considering the elements appearing in the
diagonal of the table we can define a closure operator on the cuts.
This technique can be used for any algebraic structure with one
multiplication operation. When considering transformation semigroups
in particular, the automorphism group can be used to find new complete
cuts. For a computational implementation, permutation groups provide
convenient test cases for the algorithm since for groups the number of
subgroups and conjugacy classes are easy to calculate.
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