Groups acting on trees and the Eilenberg-Ganea problem for families

TL;DR

This paper constructs new examples of groups with specific cohomological and geometric dimensions related to various subgroup families, using Bass-Serre theory and previous examples.

Contribution

It introduces new groups with cohomological dimension 2 and geometric dimension 3 for multiple subgroup families, advancing understanding of the Eilenberg-Ganea problem.

Findings

Groups with cohomological dimension 2 and geometric dimension 3 for finite subgroups

Groups with cohomological dimension 2 and geometric dimension 3 for virtually abelian groups

Groups with cohomological dimension 2 and geometric dimension 3 for virtually poly-cyclic subgroups

Abstract

We construct new examples of groups with cohomological dimension 2 and geometric dimension 3 with respect to the families of finite subgroups, virtually abelian groups of bounded rank, and the family of virtually poly-cyclic subgroups. Our main ingredients are the examples constructed by Brady-Leary-Nucinckis and Fluch-Leary, and Bass-Serre theory.