# Eighth Barcelona Weekend in Group Theory (2013)

**Friday April 26th, Aula A1, CRM, Bellaterra**

- 15:30
**Delaram Kahrobaei**(CUNY, New York)*Public key exchange using semidirect product of (semi)groups.*In this talk, I describe a brand new key exchange protocol based on a semidirect product of (semi)groups (more specifically, an extension of a (semi)group by automorphisms), and then focus on practical instances of this general idea. Our protocol can be based on any group, in particular on any non-commutative group. One of its special cases is the standard Diffie-Hellman protocol, which is based on a cyclic group. However, when our protocol is used with a non-commutative (semi)group, it acquires several useful features that make it compare favorably to the Diffie-Hellman protocol. Here we also suggest a particular non-commutative semigroup (of matrices) as the platform and show that security of the relevant protocol is based on a quite different assumption compared to that of the standard Diffie-Hellman protocol. This is a joint work with M.Habeeb, C.Koupparis and V.Shpilrain.

- 16:45
**Andrzej Zuk**(Université Paris 7)*Random walks on symmetric groups.* - 18:00
**Jon González**(Universidad Autónoma de Madrid)*The orbit method.*Around late 50s and early 60s, Alexandre Kirillov developed the orbit method to study the unitary representations of simply connected nilpotent Lie groups. If G is a simply connected nilpotent Lie group, this method provides a natural correspondence between the equivalence classes of unitary irreducible representations of G and the orbits of the action of G in the dual space of its Lie algebra.

It turned out that the orbit method works in more general situations (for example, for solvable or simple Lie groups) and even beyond the realm of Lie groups. Khazdan proved that the orbit method also works in the case of finite p-groups of nilpotency class smaller than $p$ and Howe proved the same result for uniformly powerful pro-p groups when p is an odd prime.

In this talk we will explain the orbit method for p-groups and pro-p groups and we will give some applications.

**Saturday April 27th, Aula 103, FME UPC, Pau Gargallo 5, Barcelona **

- 10:00
**Ana Khukhro**(Université de Neuchâtel, Switzerland)*Finiteness properties of non-uniform lattices.*Box spaces are an important class of metric spaces which arise from finitely generated residually finite groups. In coarse geometry, box spaces are often the only known examples of bounded geometry metric spaces possessing a given property of interest. This is partly due to the fact that there are many remarkable links between the analytic properties of a group and the geometric properties of the associated box space, since it allows us to control these spaces using information about the group. I will give an overview of the subject and describe some recent progress.

- 11:15
**Francesco Matucci**(Université Paris-Sud 11)*Measuring finiteness in groups.*Given a residually finite group, we analyze a growth function measuring the minimal index of a normal subgroup in a group which does not contain a given element. This growth (called Farb growth) attempts to measure how ''effective'' a group is residually finite. By a result of Bou-Rabee and McReynolds, the Farb growth function can be used to give a Gromov-like characterization of virtually nilpotent groups in a particular case. We study the case of free groups and recover and improve bounds on the Farb growth by rephrasing the problem in terms of laws of a group. The investigation naturally leads to the study of a second related growth function which measures asymptotic behavior of the index of the intersection of all subgroups of a group that have index at most n and which gives estimates about the previous growth function. In this talk I will introduce these growths and give an overview of some cases and properties. This is joint work with Ian Biringer, Khalid Bou-Rabee and Martin Kassabov.

- 12:30
**Ariadna Fossas**(EPFL, Lausanne, Switzerland)*Stasheff's associahedra and Thompson's groups.*