Unitary representations and Kazhdan's property (T)

Details

Supervisors

Caprace, Pierre-Emmanuel

Faculty

Faculté des sciences

Degree label

Master [120] en sciences mathématiques, à finalité approfondie

Abstract

Kazhdan's property (T) is a rigidity property of unitary representations of groups. More precisely, we say that a group has property (T) if, whenever a unitary representation almost has invariant vectors, then it has non-trivial invariant vectors. Originally introduced by Kazhdan as a tool to study lattices in real Lie groups, property (T) revealed to be a fundamental concept in the study of locally compact groups. In this thesis, we introduce Kazhdan's property for discrete groups, and investigate some important consequences on the structure of groups, such as finite generation. On the other hand, identifying which finitely generated groups have property (T) is an interesting problem. All finite groups have property (T), but it is relatively difficult to give examples of infinite groups with property (T). This motivates the development of criteria for property (T). In particular, a geometric criterion involving a quantity called the representation angle between subgroups provides a powerful method which can yield many examples of infinite groups with property (T). Following this approach, we present the result of some original research, thereby yielding new examples of infinite hyperbolic groups with property (T).