# Seventh Barcelona Weekend in Group Theory (2012)

Speakers: Yago Antolín Pichel (University of Southampton), Laura Ciobanu (Université de Fribourg), Giovanni Gandini (Hausdorff Center for Mathematics, Bonn), Aditi Kar (Oxford University), Cyril Nicaud (LIGM, Université Paris-Sud).

**Friday May 4th, Aula A1, CRM, Bellaterra**

- 15:00
**Laura Ciobanu**(Université de Fribourg)*Conjugacy growth series and languages in groups.*

I will mention a few facts on the standard conjugacy growth in groups, and show that the language S of shortlex conjugacy representatives has some unexpected behaviour in free groups and free products. I will then introduce the geodesic conjugacy language L and geodesic conjugacy growth series A for a finitely generated group, and present the interactions between rationality of both the geodesic conjugacy growth series and standard conjugacy growth series, as well as regularity of the S and L, with various group constructions. This is joint work with Susan Hermiller. - 16:15
**Aditi Kar**(Oxford University)*One relator quotients of graph products.*

I will be talking about my joint paper with Yago Antolin of the same name as the title. In this paper, we generalise Magnus' Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups. - 17:30
**Giovanni Gandini**(Hausdorff Center for Mathematics, Bonn)*Finiteness properties of non-uniform lattices.*

We show that a non-uniform lattice in the automorphism group of a locally finite n-dimensional contractible CW-complex is not of type FP_n. In dimension two it solves a conjecture of Farb, Hruska and Thomas. Moreover, it gives an easier proof of a result of Bux and Wortman on S-arithmetic groups over function fields that solved a long-standing conjecture.

**Saturday May 5th, Aula 001, FME UPC, Pau Gargallo 5, Barcelona**

- 10:30
**Yago Antolín Pichel**(University of Southampton)*Geodesic growth in right-angled and even Coxeter groups*

It has long been known that the spherical or standard growth of a right-angled Coxeter (or Artin) group depends only on the f-polynomial of the graph it is based on. Thus there are many non-isomorphic right- angled Coxeter (or Artin) groups with the same spherical growth. In this talk we consider geodesic instead of spherical growth, and discuss which combinatorial properties of a regular graph can completely determine the geodesic growth of the right-angled Coxeter group this graph defines. As a consequence, we provide the first known examples of right-angled and even Coxeter groups with the same geodesic growth series. This is joint work with Laura Ciobanu. - 11:45
**Cyril Nicaud**(LIGM, Université Paris-Sud)*Statistical Properties of Subgroups of Free Groups.*

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated by randomly and uniformly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In the first part of the talk, we extend the model for random tuples of various size (k is not fixed anymore) and for reduced words that are generated by a Markovian automaton. We show by probabilistic arguments that under this rather relaxed assumptions, the good properties of the word-based distribution are preserved. In the second part of the talk we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution. Here, subgroups are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these distribution behave quite differently, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. These are joint work with Frédérique Bassino, Armando Martino, Enric Ventura and Pascal Weil.