Normal subgroups of SimpHAtic groups

Author:

Osajda Damian12

Affiliation:

1. Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50–384 Wrocław, Poland

2. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

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 non-uniform 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. In Appendix we present the topological two-dimensional quasi-Helly property of systolic complexes.

Publisher

World Scientific Pub Co Pte Ltd

Subject

Geometry and Topology,Analysis

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