Spherical p-group complexes arising from finite groups of Lie type

Piterman, Kevin Ivan

doi:10.2140/agt.2026.26.791

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Spherical $p$-group complexes arising from finite groups of Lie type

Kevin Ivan Piterman

Algebraic & Geometric Topology 26 (2026) 791–824

DOI: 10.2140/agt.2026.26.791

Abstract

We show that the $p$ -group complex of a finite group $G$ is homotopy equivalent to a wedge of spheres of dimension at most $n$ if $G$ contains a self-centralising normal subgroup $H$ which is isomorphic to a group of Lie type and Lie rank $n$ in characteristic $p$ . If in addition, every order- $p$ element of $G$ induces an inner or field automorphism on $H$ , the $p$ -group complex of $G$ is $G$ -homotopy equivalent to a spherical complex obtained from the Tits building of $H$ .

We also prove that the reduced Euler characteristic of the $p$ -group complex of a finite group $G$ is nonzero if $G$ has trivial $p$ -core and $H$ is a self-centralising normal subgroup of $G$ which is a group of Lie type (in any characteristic), except possibly when $p = 2$ and $H = A_{n} (4^{a})$ ( $n \geq 2$ ) or $E_{6} (4^{a})$ . In particular, we conclude that the Euler characteristic of the $p$ -group complex of an almost simple group does not vanish for $p \geq 7$ .

Keywords

$p$-subgroup complexes, Tits buildings, finite groups of Lie type

Mathematical Subject Classification

Primary: 06A11, 20E42, 55U05

Secondary: 05E18, 20D30, 20G40

References

Publication

Received: 15 November 2024

Revised: 14 February 2025

Accepted: 15 March 2025

Published: 11 February 2026

Authors

Kevin Ivan Piterman
Fachbereich Mathematik und Informatik Philipps-Universität Marburg Marburg Germany	Department of Mathematics and Data Science Vrije Universiteit Brussel Brussels Belgium

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Volume 26, issue 2 (2026)