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

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 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 A 5 in dimensions 2, 3 and 5, the linear fractional group PSL 2 ( 7 ) in dimension 5, and possibly the unitary group 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.