Number Theory Seminar
An arithmetic group is a group commensurable with the integral points of a linear algebraic group defined over the rationals. A classical theorem of Borel and Harish-Chandra asserts that such groups are finitely presented; their proof was turned into an algorithm by Grunwald and Segal, but without a complexity analysis of the corresponding algorithm. In the recent years, deeper invariants of arithmetic groups have attracted a lot of interest: their cohomology groups, equipped with the action of Hecke operators. I will present ongoing work with Michael Lipnowski: we describe an algorithm which, given an arithmetic group Gamma whose corresponding arithmetic manifold is compact, computes the cohomology of Gamma together with the action of Hecke operators, and we analyse its complexity.