On minimal actions of finite simple groups on homology spheres and Euclidean spaces

TL;DR

This paper investigates when finite simple groups have the same minimal dimension for faithful smooth actions on homology spheres and faithful linear actions on spheres, providing new results for specific groups.

Contribution

It establishes that for certain groups like PSL(2,p) and some alternating and symmetric groups, the minimal dimensions for smooth and linear actions are equal, extending to Euclidean spaces.

Findings

Minimal dimensions coincide for PSL(2,p) groups.

Minimal dimensions coincide for certain alternating and symmetric groups.

Results extend to actions on Euclidean spaces.

Abstract

We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2,p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.