Arithmetic groups and their profinite completions

17 September - 17 December 2026

Thursdays, 13:15 - 15:00

Location: HG G 43

First lecture: 17 September

Abstract

The main aim of the course will be to get to proofs of both positive and negative results in the setting of the question of whether arithmeticgroups are profinitely rigid or not (a topic that has received significant attention of late): in basic terms whether (arithmetic) groups have the same set of finite quotients. The positive results will be in the context of arithmetic Fuchsian and Kleinian groups, and the negative results in the setting of the Congruence Subgroup Property. Part of the course will be dedicated to a detailed exposition ofarithmetic Fuchsian and Kleinian groups and the geometric and topological properties of the associated quotient hyperbolic orbifolds. The aforementioned positive profinite rigidity resultsrequire a very precise understanding of the space of SL(2,C) representations, and the underlyingarithmetic constructions via quaternion algebras over number fields.