The cohomological restriction map and FP-infinity groups

TL;DR

This paper investigates the properties of the cohomological restriction map in groups, establishing conditions under which its kernel is finitely generated, especially focusing on groups with virtual finite cohomological dimension.

Contribution

It proves that for groups with virtual finite cohomological dimension, the kernel of the restriction map is finitely generated, and shows counterexamples when this condition is not met.

Findings

Kernel is finitely generated for groups with virtual finite cohomological dimension

Counterexamples exist for groups without this property, even if they are FP-infinity groups

Provides conditions under which the cohomological restriction map behaves finitely

Abstract

We ask, following Bartholdi, whether it is true that the kernel of the restriction map from the cohomology of a group G to the cohomology of a finite index subgroup H is finitely generated as an ideal. We show that in case the group has virtual finite cohomological dimension it is true, and we will show that if G does not have virtual finite cohomological dimension it might not be true, even in case G is an FP infinity group.