On fixed point sets and Lefschetz modules for sporadic simple groups

TL;DR

This paper investigates the fixed point sets and Lefschetz modules associated with sporadic simple groups, revealing homotopy equivalences and subgroup structures for various primes and group actions.

Contribution

It provides new computations of fixed point sets and describes vertices of Lefschetz modules for sporadic groups, linking subgroup complexes to geometric structures.

Findings

Fixed point sets for six sporadic groups are homotopy equivalent to standard geometries.

Computed fixed point sets for sporadic groups with extraspecial Sylow p-subgroups.

Described vertices for summands of Lefschetz modules associated with these groups.

Abstract

We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component of the centralizer. For odd primes, fixed point sets are computed for sporadic groups having an extraspecial Sylow p-subgroup of order p^3, acting on the complex of those p-radical subgroups containing a p-central element in their centers. Vertices for summands of the associated reduced Lefschetz modules are described.