Schedule

A hyperbolic Coxeter group $G$ is a cofinite discrete group generated by a finite set $S$ of hyperplane reflections in hyperbolic space $\mathbb H^n$.

The growth series $f_S(t)$ of $G$ is the formal power series whose coefficients $a_k$ count the number of words of $S$-length equal to $k$. The growth rate of $G$ as given by the inverse of the convergence radius of $f_S(t)$ turns out to be an algebraic integer >1.

By a result of E. Hironaka, the smallest growth rate of any cocompact planar hyperbolic Coxeter group is achieved by the Coxeter triangle group (2,3,7) and equals Lehmer’s number $\alpha_L\approx1.17628$.

The number $\alpha_L$ is the smallest known Salem number and of particular importance in view of Lehmer’s Conjecture.

Beyond that, and by a result of Siegel, the Coxeter group (2,3,7) is also distinguished by realising minimal covolume among all fundamental groups of hyperbolic orbifolds of dimension two.

We shall discuss similar connections and related aspects from various point of views and especially in higher dimensions $n>2$.  In particular, we will present our result stating that NOT every Salem number appears as the growth rate of a cocompact hyperbolic Coxeter group (joint with L. Liechti).

In his work on the Tarski conjecture Sela proved, among other things, that up to fractional Dehn twists, there is a uniform upper bound on the number of ways to write a fixed commutator c as [u,v] in a free group. I will give an explicit upper bound. This is joint work with Aminaei, Conway, Florescu, Kim, and Wilton.

Simple algebraic structures, ranging from groups to Lie and associative algebras, can often be generated rapidly using just a few elements. In this talk, we will explore results addressing key questions such as:

1. Is the structure generated by two elements? (2-generation)

2. Do randomly selected elements generate the structure? (random generation)

3. Can every element be expressed as a short product or sum of certain special elements? (Waring-type problems)

4. Is the generation process in the previous cases efficient? (rapid generation)

When working over a group instead of a field, the analogue of a polynomial is just a word w(x_1, ..., x_n) which can involve constants from the group. Following this analogy, one can study varieties, and then algebraic groups (varieties endowed with a group law which can be expressed by such a "polynomial").

Over the free group, varieties are well understood since the works of Makanin-Razborov and later Sela. In a joint work with Guirardel, we show that there are very few irreducible algebraic groups over the free group, and we describe all such structures. One of the key tools in this work is JSJ decomposition of groups, which enables us to give a description of automorphisms of the coordinate group of such a variety in terms of automorphisms of fundamental groups with boundaries.

In a recent paper, Lucchini and Thakkar present a deterministic algorithm

for computing a smallest sized generating set of a finite group G. The

complexity is polynomial in |G| and, for a subgroup of the symmetric group

Sym(n), it is polynomial in n and the size of the largest nonabelian

composition factor of G.

We describe a randomized version of the algorithm that has expected time

polynomial in n for a subgroup of Sym(n). This is joint work with Gareth

Tracey. More precisely, the expected number of generating tests required by

our algorithm is O(n^2 log n), where a generating test consists of checking

whether a given subset of G generates G. The Schreier-Sims algorithm, which

has running time polynomial in n, can be used for this purpose.

We have implemented our algorithm in Magma, and it runs very fast in practice

and it can, for example, quickly verify the well-known fact that A_5^{19} and

A_5^{20} are respectively 2- and 3-generated groups.

Another recent result (which is related, although not obviously so) is that,

if G is a subgroup of S_n and N is a minimal normal nonabelian subgroup of G, then G/N embeds in S_n. 

The notion of a binary permutation group​ was introduced by Gregory Cherlin, who sought to use model theory to understand sporadic behaviour in the universe of finite permutation groups. 

A rough definition goes as follows: Let G be a finite permutation group acting on a set X. The action of G on X extends in an obvious way to an action on X^2, the Cartesian product of X with itself and, more generally, to an action on X^n, for n any positive integer. We call G a binary permutation group​ if, for any positive integer n, the orbits of G on X^n can be deduced directly from the orbits of G on X^2.

In this talk we discuss recent progress in classifying the almost simple​ binary permutation groups. In particular we will introduce a natural family of graphs whose vertices are elements of an almost simple group G and we will show how the connectedness (or otherwise) of this graph can be used to understand the binary actions of G.

This is joint work with Pierre Guillot (Strasbourg, France), Martin Liebeck (Imperial, UK) and Pablo Spiga (Milano-Bicocca, Italy).

The lower central series of a group $G$ is defined by $\gamma_1=G$ and $\gamma_n = [G,\gamma_{n-1}]$. The "dimension series", introduced by Magnus, is defined using the group algebra over the integers: $\delta_n = \{g: g-1\text{ belongs to the $n$-th power of the augmentation ideal}\}$.

It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has $\delta_n\ge\gamma_n$, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with $\delta_4/\gamma_4$ cyclic of order 2. On the positive side, Sjogren showed that $\delta_n/\gamma_n$ is always a torsion group, of exponent bounded by a function of $n$. Furthermore, it was believed (and falsely proven by Gupta) that only $2$-torsion may occur.

In joint work with Roman Mikhailov, we prove however that every torsion abelian group may occur as a quotient $\delta_n/\gamma_n$; this proves that Sjogren's result is essentially optimal.

Even more interestingly, we show that this problem is intimately connected to the homotopy groups $\pi_n^(S^m)$ of spheres; more precisely, the quotient $\delta_n/gamma_n$ is related to the difference between homotopy and homology. We may explicitly produce $p$-torsion elements starting from the order-$p$ element in the homotopy group $\pi_{2p}(S^2)$ due to Serre.