Normal subgroups of SimpHAtic groups

TL;DR

This paper investigates the structure of normal subgroups within SimpHAtic groups, revealing they are either finite, of finite index, or virtually free, and explores related properties of group actions and extensions.

Contribution

It establishes strong restrictions on normal subgroups of SimpHAtic groups and related classes, extending understanding of their algebraic and geometric properties.

Findings

Normal subgroups are finite, of finite index, or virtually free.

Nonuniform lattices in SimpHAtic complexes are not finitely presentable.

Finitely presented groups acting properly on these complexes act geometrically on SimpHAtic complexes.

Abstract

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index, or virtually free. This result applies, in particular, to normal subgroups of systolic groups. We prove similar strong restrictions on group extensions for other classes of asymptotically aspherical groups. The proof relies on studying homotopy types at infinity of groups in question. We also show that nonuniform lattices in SimpHAtic complexes (and in more general complexes) are not finitely presentable and that finitely presented groups acting properly on such complexes act geometrically on SimpHAtic complexes.