November 2017 abstracts

Speaker: Laura Ciobanu (Heriot-Watt)
Title: Cayley graphs of relatively hyperbolic groups and formal languages
Abstract: In this talk I will show how given a finitely generated relatively hyperbolic group G, one can construct a finite generating set X of G for which (G,X) has a number of nice metric properties, provided that the parabolic subgroups have these properties.

I will discuss the applications of these properties to the growth series, language of geodesics, biautomatic structures and conjugacy problem. This is joint work with Yago Antolin.

Speaker: James Mitchell (St Andrews)
Title: Universal sequences for groups and semigroups
Abstract: Ore’s Theorem from 1951 states that every element of the symmetric group
S_X on an infinite set X is a commutator. In other words, for any
permutation p on an infinite set X, the equation p = a^{-1}b^{-1} ab has a solution in the symmetric group S_X.  If w is any word over a
finite alphabet that is not a proper power of another word, and p is any
permutation of an infinite set X, then Silberger, Droste, Dougherty,
Mycielski, and Lyndon showed that p = w has a solution in permutations of
X.

A universal sequence for a group or semigroup G is a sequence of
words w_1, w_2, \ldots such that for any sequence g_1, g_2, \ldots\in G
the equations w_i = g_i, i\in \mathbb{N}, can be solved
simultaneously in G. Galvin showed that the sequence
\{a^{-1}(a^iba^{-i})b^{-1}(a^ib^{-1}a^{-i})ba:i\in\mathbb{N}\} is universal
for the symmetric group S_X when X is infinite. On the other hand, if G
is any countable group, then G has no universal sequences.

In this talk, I will discuss properties of universal sequences for some
well-known groups and semigroups.

Speaker: Jarek Kedra (Aberdeen)
Title: Conjugation invariant geometry of {\rm SL}(n,{\mathbb Z})
Abstract: It is known that the group {\rm SL}(n,{\mathbb Z}) for n>2 is boundedly generated by elementary matrices. It almost immediately follows that every conjugation invariant norm on {\rm SL}(n,{\mathbb Z}) is bounded. In particular, the word norm associated with a conjugation invariant generating set has finite diameter. I will discuss the dependence of the diameter on the choice of a generating set and present applications to the finite simple groups {\rm PSL}(n,q).
This is recent joint work with Assaf Libman and Ben Martin.

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