Piecewise linear actions and Zimmer's program

TL;DR

This paper investigates Zimmer's program for lattice actions on surfaces via piecewise linear homomorphisms, showing that for most surfaces, such actions factor through finite groups, extending stability results to PL settings.

Contribution

It introduces a PL version of Reeb-Thurston's stability and proves that lattice actions on certain surfaces are finite, advancing Zimmer's program in the PL context.

Findings

Actions on non-torus, Klein bottle surfaces are finite.

PL Reeb-Thurston stability is established.

Lattice actions factor through finite groups for n>4.

Abstract

We consider Zimmer's program of lattice actions on surfaces by PL homomorphisms. It is proved that when the surface is not the torus or Klein bottle the action of any finite-index subgroup of SL(n,Z), n>4, (more generally for any 2-big lattice), factors through a finite group action. The proof is based on an establishment of a PL version of Reeb-Thurston's stability.