**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

on an infinite set is a commutator. In other words, for any

permutation on an infinite set , the equation has a solution in the symmetric group . If is any word over a

finite alphabet that is not a proper power of another word, and is any

permutation of an infinite set , then Silberger, Droste, Dougherty,

Mycielski, and Lyndon showed that has a solution in permutations of

.

A *universal sequence* for a group or semigroup *G* is a sequence of

words such that for any sequence

the equations , , can be solved

simultaneously in . Galvin showed that the sequence

is universal

for the symmetric group when is infinite. On the other hand, if

is any countable group, then 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

**Abstract: **It is known that the group for is boundedly generated by elementary matrices. It almost immediately follows that every conjugation invariant norm on 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 .

This is recent joint work with Assaf Libman and Ben Martin.