Effective Methods for Arithmetic Groups

Arithmetic groups are groups that can be written as the set of all matrices with integer entries in a suitable linear group.

They are objects of fundamental interest in many areas of mathematics, including geometry, topology and number theory.

The aim of this course is to develop tools to find a fundamental domain for the action of a given arithmetic group of isometries of a (real or complex) hyperbolic space.

This will allow us to write explicit finite presentations, to solve the word problem, and to get a list of conjugacy classes of finite subgroups, for instance.