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

2024/03/12 by Piterman, Kevin Iván · 1 citation
#05E18 #06A11 #20D30 #20E42 #20G40 #55U05 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2403.07489

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 non-zero 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≥ 2) or E₆(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≥ 7.

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• Components in characteristic p and Quillen's conjecture

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