Abstracts

We present a new procedure to determine the growth function of an Artin-Tits monoid of spherical type (hence of a braid monoid) with respect to the standard generators, as the inverse of the determinant of a very simple matrix.

Using this approach, we show that the exponential growth rates of the positive braid monoids $A_n$ tend to 3.233636… as $n$ tends to infinity. This number is well-known, as it is the growth rate of the coefficients of the only solution $x_0(y)=−(1+y+2y^2+4y^3+9y^4+⋯)$ to the classical partial theta function $\sum_{k=0}^{\infty}{y^{k\choose 2} x^k}$.

We also describe the sequence $1,1,2,4,9,\ldots$ formed by the coefficients of $−x_0(y)$, by showing that its $k$th term (the coefficient of $y_k$) is equal to the number of braids of length $k$, in the positive braid monoid $A_{\infty}$ on an infinite number of strands, whose maximal lexicographic representative starts with the first generator $a_1$. This is an unexpected connection between the partial theta function and the theory of braids.

Joint with Ramón J. Flores.

Gromov's monsters are finitely generated groups whose Cayley graphs contain expander graphs in a reasonable geometric sense - a property of great interest in geometric and analytic group theory.

I will discuss the two known constructions of such groups and explain that they yield two classes of groups that are very different with respect to their actions on and geometric features of Gromov hyperbolic spaces.

Based on joint works with Alessandro Sisto and Romain Tessera.

Let G be a finitely generated group. A subgroup H is separable if it is closed in the profinite topology; equivalently, for each element g of G-H, there is a finite index subgroup G' of G that contains H but not g. In particular, G is residually finite if the subgroup {1} is separable. A natural question is: given g, what is the index (in terms of the word length of g, and some finite amount of data about H) of a lowest-index G' containing H but not g?This is quantified by the "separability growth function" of the pair (G,H); when H={1}, this is the "residual finiteness growth" function of G, introduced by Bou-Rabee in 2010. The growth rates of these functions are independent of the choice of generating set.

The problem of estimating these functions often has an algebraic flavour, when trying to compute lower bounds, but estimates from above can sometimes be made by geometric/topological considerations. For example, classical techniques due to Stallings allow one to make such estimates when G is free and H is finitely generated, by considering covering spaces of graphs. I will discuss how these techniques can be extended to provide upper bounds in the case where G is the fundamental group of a compact special cube complex (e.g. when G is a right-angled Artin group). I will also discuss some related open questions dealing with hyperbolic CAT(0) cubical groups, e.g. fundamental groups of closed hyperbolic 3-manifolds.

This talk is on joint work with Khalid Bou-Rabee and Priyam Patel, and other joint work with Priyam Patel.

Let G be a discrete group and M a n by n matrix over the group ring K[G], where K is a subfield of the field of complex numbers. M acts as bounded operator m_M on the Hilbert space l2(G)^n extending the multiplication in C[G], where l2(G) is the space of square summable functions on G.

If G is finite, all eigenvalues of this operator are zeros of its characteristic polynomial which has coefficients in K, and so, they are algebraic over K.

We will show that if G is sofic or an one-relator group, then, as in the finite case, the eigenvalues of m_M are algebraic over K.

Part of the talk is based on a joint work with Diego López-Álvarez.

The subgroup congruence problem for an (algebraic) group G makes a statement about the form of the subgroups of finite index in G. We will start with a survey on SCP, emphasizing the case G= SL_n(D) and the role of its elementary subgroup E_n(D).

Next, we will explain the impact of the (failure to) SCP to classical questions in group rings. To end, we will present an alternative approach to the latter questions via fixed point properties on trees and amalgamated products. The last part is based on joint work with Andreas Bächle, Eric Jespers, Ann Kiefer and Doryan Temmerman.

Every action of a group on a rooted tree induces an action on the boundary of the tree. This action yields various representations of the group on spaces of functions on the boundary. In this talk we discuss the structure of such representations. After reviewing of the basic concepts we will introduce locally 2-transitive actions on trees and give examples of such actions. It will be discussed how this property gives rise to an explicit decomposition of the boundary representation into irreducible constituents.

I will discuss some questions about spectra of groups and group actions and methods to answer them, including some unexpected reduction to the theory of quasi-crystals.

Quotient varieties for reductive groups admit the Kirwan (partial) resolution of singularities, and quite often also a "noncommutative resolution". We will motivate the appearance of noncommutative resolutions (via McKay correspondence), and compare them to their commutative counterparts. This is a joint work with Michel Van den Bergh.