Finite generation properties of cohomology rings for infinite groups

Given a group G which is not necessarily finite, and a Noetherian commutative ring R, an important question is when the cohomology ring H∗(G,R) is a Noetherian ring. For finite groups, this is the Venkov–Evens theorem. This theorem is extended to finite dimensional restricted Lie algebras when R is the ground field and to more general finite groups schemes over a field R by Friedlander and Suslin. Recently, van der Kallen and Touzé have shown this for reductive algebraic groups over a field of positive characteristics with rational group cohomology. In this note, we prove a finite generation theorem for a large class of infinite groups which include all arithmetic subgroups of linear algebraic groups, lattices of semisimple Lie groups, mapping class groups of surfaces, outer automorphism groups of free groups, Gromov hyperbolic groups, outer automorphism groups of free groups, and many other natural infinite groups.