On asymptotically hereditarily aspherical groups

TL;DR

This paper systematically studies asymptotically hereditarily aspherical (AHA) groups, providing new examples, exploring their topological properties at infinity, and relating these to systolic groups and their boundaries.

Contribution

It introduces numerous new examples of AHA groups, especially in high dimensions, and connects their properties to the topology at infinity and systolic group characteristics.

Findings

Many new high-dimensional AHA group examples

AHA property relates to the topology at infinity

Boundary at infinity properties for certain AHA groups

Abstract

We undertake a systematic study of asymptotically hereditarily aspherical (AHA) groups - the class of groups introduced by Tadeusz Januszkiewicz and the second author as a tool for exhibiting exotic properties of systolic groups. We provide many new examples of AHA groups, also in high dimensions. We relate AHA property with the topology at infinity of a group, and deduce in this way some new properties of (weakly) systolic groups. We also exhibit an interesting property of boundary at infinity for few classes of AHA groups.