A note on actions of the symplectic group Sp(2g,Z) on homology spheres

TL;DR

The paper proves that for symplectic groups Sp(2g,Z), any continuous action on homology spheres of dimension less than 2g-1 is trivial, extending known results for linear groups.

Contribution

It generalizes previous results by showing triviality of actions of Sp(2g,Z) on low-dimensional homology spheres, including spheres, for g > 2.

Findings

Actions of Sp(2g,Z) on homology spheres of dimension less than 2g-1 are trivial.

Any continuous action of Sp(2g,Z) on S^m with m < 2g-1 is trivial.

The result extends known linear group actions to symplectic groups.

Abstract

The symplectic group Sp(2g,Z) is a subgroup of the linear group SL(2g,Z) and admits a faithful action on the sphere S^(2g-1), induced from its linear action on Euclidean space R^(2g). Generalizing corresponding results for linear groups, we show that, if m < 2g-1 and g > 2, any continuous action of Sp(2g,Z) on a homology m-sphere, and in particular on S^m, is trivial.