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
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.