# 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.