Automorphisms and Cohomology of Discrete Groups

Abstract

Let G denote a discrete group of finite cohomological dimension. Calculating the cohomology of such groups is notoriously difficult, involving complicated geometric information attached to the group. For example, the cohomology of torsion-free arithmetic groups involves delicate questions about symmetric spaces and number theory. Perhaps the key difficulty lies in that there is no practical method for building up the Ž cohomology of G from that of its subgroups such as one can do for finite . groups . In this paper we outline a method for constructing non-trivial classes in U Ž . Ž . H G, F where F denotes a field with p elements based on the use of p p finite automorphism groups of G. Given an explicit presentation for G, Ž . Aut G can often be approached, and in particular its finite subgroups are sometimes accessible. Furthermore, it is an elementary observation that many interesting classes of groups admit numerous finite symmetries. An obvious but very important example is given by a normal torsion-free subgroup G in an arithmetic group U; any finite subgroup K : U will act on it via conjugation. Ž . Given G : Aut G , we can form the semidirect product G s G = G. w Let H : G denote a finite subgroup mapping onto G under the natural Ž . Ž . projection map. Then C H s C H l G : G and in the particular case G G Ž . < .4 when H s 1, x x g G , we have that