On finite simple groups acting on homology spheres with small fixed point sets

Abstract

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets (“pseudofree action”) is the alternating group \(\mathbb {A}_5\) acting on the 2-sphere. Our first main theorem is the finiteness result that there are only finitely many finite simple groups which admit a smooth action on a homology sphere with at most \(d\)-dimensional fixed points sets, for a fixed \(d\). We then go on proving that the finite simple groups acting on a homology sphere with at most 1-dimensional fixed point sets are the alternating group \(\mathbb {A}_5\) in dimensions 2, 3 and 5, the linear fractional group \(\mathrm{PSL}_2(7)\) in dimension 5, and possibly the unitary group \(\mathrm{PSU}_3(3)\) in dimension 5 (we conjecture that it does not admit any action on a homology 5-sphere but cannot exclude it at present). Finally, we discuss the situation for arbitrary finite groups which admit an action on a homology 3-sphere.