The quasi-isometry invariance of the Coset Intersection Complex

TL;DR

The paper introduces the coset intersection complex for group pairs, proving its quasi-isometry and homotopy invariance, and connects algebraic subgroup properties to topological features, with applications to right-angled Artin groups.

Contribution

It defines the coset intersection complex and establishes its invariance under quasi-isometry, linking algebraic subgroup properties to topological invariants.

Findings

The coset intersection complex is a quasi-isometry invariant of group pairs.

Certain algebraic properties of subgroups correspond to topological features of the complex.

For specific right-angled Artin groups, the complex is quasi-isometric to a tree.

Abstract

For a pair $(G,\mathcal{P})$ consisting of a group and finite collection of subgroups, we introduce a simplicial $G$-complex $\mathcal{K}(G,\mathcal{P})$ called the coset intersection complex. We prove that the quasi-isometry type and the homotopy type of $\mathcal{K}(G,\mathcal{P})$ are quasi-isometric invariants of the group pair $(G,\mathcal{P})$. Classical properties of $\mathcal{P}$ in $G$ correspond to topological or geometric properties of $\mathcal{K}(G,\mathcal{P})$, such as having finite height, having finite width, being almost malnormal, admiting a malnormal core, or having thickness of order one. As applications, we obtain that a number of algebraic properties of $\mathcal{P}$ in $G$ are quasi-isometry invariants of the pair $(G,\mathcal{P})$. For a certain class of right-angled Artin groups and their maximal parabolic subgroups, we show that the complex $\mathcal{K}(G,\mathcal{P})$ is quasi-isometric to the Extension graph; in particular, it is quasi-isometric to a tree.