Trees, homology, and automorphism groups of RAAGs

TL;DR

This paper investigates the homology of a specific subgroup of automorphisms of right-angled Artin groups defined by trees, providing bounds and average case analysis based on graph properties.

Contribution

It introduces a lower bound on the first Betti number for these subgroups and analyzes its average value using combinatorial methods.

Findings

Lower bound on the first Betti number based on deep vertices.

Average Betti number depends on the size and structure of the defining tree.

Combinatorial analysis links graph properties to homological invariants.

Abstract

We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this subgroup, based on the number and degree of a certain type of vertices, which we call deep. We then use combinatorial methods to analyze the average value of this Betti number, in terms of the size of the defining tree.