Relative ends and splittings of groups

Abstract

This thesis is motivated by a long-standing conjecture on groups with Bredon cohomological dimension one and their action on trees with stabilisers in a specific family of subgroups. Chapter 1 consists of the first approach to deal with the problem following steps of known results for families of finite and virtually cyclic subgroups. As a consequent of this attempt, we answer a question on the Bredon cohomological and geometric dimension of free abelian groups with finite rank.

The Main Theorem in Chapter 2 provides a partial answer to Kropholler’s Conjecture on splittings of groups, which has been thought to be an alternative step for the proof of the conjecture stated in Chapter 1. We define the notion of relative ends, commensurable subgroups, almost invariant sets and the relation between those and splittings of groups, or equivalently, actions on trees with special stabilisers.

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ORCID for Nansen Petrosyan: orcid.org/0000-0002-2768-5279

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