Ph.D Thesis
Ph.D Student Ehud Meir Ben Efraim On Certain Cohomological Properties of a Group and their Reflection in a Finite Index Subgroup Department of Mathematics Full Professor Aljadeff Eli Full Professor Gelaki Shlomo

Abstract

This thesis deals with relations between cohomological properties of a group and a finite index subgroup.

The first chapter deals with a conjecture of Moore. Moore's conjecture states that if G is a group and H a finite index subgroup of G, then under a certain natural assumption on the embedding of H in G, every G module which is projective over H is projective over G. Moore's conjecture generalizes a theorem of Serre about cohomological dimension of torsion free groups. It also generalizes Chouinard's theorem for finite groups, which states that if G is a finite group, then a G module M which is projective over every elementary abelian subgroup of G is projective over G. It was known that the conjecture is true for finite groups and groups of finite cohomlogical dimension. In this thesis we generalize these results and prove that the conjecture is true for all the groups in Kropholler's hierarchy LHF. We also show certain closure properties of the class of groups which satisfy the conjecture. We use these results to construct examples for groups which satisfy the conjecture, and whose cohomology ring is a polynomial ring.

The second chapter deals with general G modules, and not necessarily projective ones. We ask how far general modules are from being projective. In case the group G is finite and we consider G modules over a field k, the non projectivity of a module M can be measured by a numerical invariant, its complexity cG(M). The Alperin Evens Theorem states that the complexity can be calculated by restricting to elementary abelian subgroups. That is- cG(M)=maxE(cE(M)), where E ranges over elementary abelian subgroups of G. The second chapter gives a generalization of Alperin-Evens Theorem for infinite groups and for arbitrary coefficient rings. We do so by using a construction of Wall, which enables us to measure complexity directly, and not via cohomology.

The third chapter deals with the restriction map from a group G to a finite index subgroup H. In case G is finite, it is known by a theorem of Quillen that the kernel of the restriction map from the cohomology ring of G to that of H is finitely generated. We ask if this finiteness condition is true also for infinite groups. We show that this is true if we also assume a strong finiteness assumption on G, and we give a general way to construct counterexamples when this assumption does not hold.