Geometry and Group Theory Ical Atom

There are two interesting norms on free groups and surface groups which are invariant under the group of all automorphisms:

A) For free groups we have the primitive norm, i.e., |g|_p = the minimal number of primitive elements one has to multiply to get g.

B) For fundamental group of genus g surface we have the simple curves norm, i.e., |g|_s = the minimal number of simple closed curves one need to concatenate to get g.

In our recent paper with M. Brandenbursky we prove the following dichotomy: either |g^n| is bounded or growths linearly with n. For free groups and surface groups we give an explicit characterisation of (un)bounded elements.

In two talks I will explain the idea of the proof and draw a number of consequences. The proof uses the theory of mapping class groups (i.e. Nielsen-Thurston normal form, Birman embedding) and quasimorphisms.

**Lecture 2. **

There are two interesting norms on free groups and surface groups which are invariant under the group of all automorphisms:

A) For free groups we have the primitive norm, i.e., |g|_p = the minimal number of primitive elements one has to multiply to get g.

B) For fundamental group of genus g surface we have the simple curves norm, i.e., |g|_s = the minimal number of simple closed curves one need to concatenate to get g.

In our recent paper with M. Brandenbursky we prove the following dichotomy: either |g^n| is bounded or growths linearly with n. For free groups and surface groups we give an explicit characterization of (un)bounded elements.

In two talks I will explain the idea of the proof and draw a number of consequences. The proof uses the theory of mapping class groups (i.e. Nielsen-Thurston normal form, Birman embedding) and quasimorphisms.

Let G be a semisimple Lie group of rank two or higher. We discuss certain asymptotic properties for sequences of lattices inside G.

A lattice in G is associated to a classical geometric object of the form M = K\G/Gamma. We allow G to be either real or p-adic. An important geometric property for such sequences of metric spaces is Benjamini-Schramm (BS) convergence. We present a theorem saying that any sequence of distinct M’s is BS-convergent.

It turns out that the geometric notion of BS-convergence has implications to representation theory, in terms of Plancherel measure convergence, and to topology, in terms of convergence of normalized Betti numbers. We will briefly mention these implications.

Kazhdan’s property (T) plays an important role in the above results. We will explain a novel approach relying on Selberg’s property instead and extending to products of rank one groups (such as SL2xSL2).

The talk is based on [Abert-Bergeron-Biringer-Gelander-Nikolov-Raimbault-Samet] and two recent preprints by Gelander-L. and L.

Bekka has defined a notion of property (T) for C-algebras that is congruous with property (T) groups in the sense that any C-completion of the complex group ring has property (T) if and only if the group does. We will outline the fairly straightforward proof of this fact, as well as discuss some ideas that Joav Orovitz and I have for “generalizing” property (T) for C*-algebras so that the analogous statement is true for groupoids and completions of their convolution algebras.