Computational Aspects of Thin Groups

Overview

Arithmetic groups have long been of interest in number theory (e.g. via the theory of automorphic forms), geometry (e.g. building interesting examples of Riemannian manifolds), and topology (e.g. computing (co)homology of locally symmetric spaces). Yet, their subgroup structure, particularly of their infinite index subgroups, remains mysterious.

Thin groups are such infinite index subgroups that have come to prominence recently because they arise naturally in various contexts with connections to number theory, geometry and topology.

The focus of our proposed 2-week event is on computational aspects of thin groups. The recent activity highlights the need for algorithms and procedures to study these groups. Our event is the first aimed specifically at addressing this topic.

We propose a 2-week event. The first week will consist of four lectures series, each consisting of 3 lectures on topics reflecting the themes of the event:

• Decidability questions for integral matrix groups.
• Computational aspects of strong approximation.
• Integral packings, reflection groups and arithmeticity.
• Computational aspects of thin groups in geometry.

These lectures are intended for graduate students and Post-Doctoral Fellows. The lecture series will be supplemented with problem sessions (with “TA support”). The second week will consist of a conference featuring leading researchers reporting on recent developments. Ample time will be set aside for informal discussions and collaborations.