Abstracts of Talks:

• 4:30 - 5:30, Friday 7th November,  Alessandra Iozzi (ETH Zurich):  

• Representations of 3-manifold groups into SL(n,C):

Let M be a 3-manifold.  A finite volume hyperbolization corresponds to a lattice injection of the fundamental group into SL(2,C) and we can think of the volume of the manifold as an invariant of this injection.  We extend this invariant to arbitrary representations of the fundamental group into SL(n,C) and we characterize the representations that correspond to its maximal value.  Our definition of the invariant depends of the study of the bounded cohomology of SL(n,C) in degree three, but the invariant has been previously considered for a more restricted family of representations by various authors using ideal triangulations of the manifold. 

• 9:30 - 10:30, Saturday 8th November,  Aditi Kar (Oxford):  

• Gradients in group theory:

Rank and Deficiency gradients quantify the asymptotics of finite approximations of a group. These group invariants have surprising connections with many different areas of mathematics: 3-manifolds, L^2 Betti numbers, topological dynamics and profinite groups.  I will give a survey of the current state of research in Gradients for groups and describe important open questions.

• 12:00 - 1:00, Saturday 8th November,  Marc Burger (ETH Zurich):   

• Lattices in products of trees and Wang's Theorem:

Groups  acting properly and co-finitely on the product of two regular trees give a class of finitely presented groups in which various behaviours are possible: there are linear examples, non-linear and SQ-universal examples, and even examples which are simple. In this talk we will first recall the general philosophy underlying our approach in the study of this class of groups; then we will treat  the question of the existence of infinite increasing towers of such groups.  This is joint work with Shahar Mozes.