Matrix group actions on product of spheres and Zimmer's program

TL;DR

This paper proves that for certain products of spheres and tori, any group action by SL(n,Z) is trivial when the number of factors is less than n-1, confirming a conjecture in Zimmer's program.

Contribution

It establishes the triviality of SL(n,Z) actions on specific manifolds, confirming a conjecture in Zimmer's program for these cases.

Findings

SL(n,Z) acts trivially on certain product manifolds when r<n-1

Confirms a conjecture in Zimmer's program for these manifolds

Provides new insights into group actions on manifolds

Abstract

Let SL(n,Z) be the special linear group over integers and $M =S^r_1 \times S^r_2,T^r_1 \times S^r_2$ , or $T^r_0 \times S^r_1 \times S^r_2$, products of spheres and tori. We prove that any group action of SL(n,Z) on $M^r$ by diffeomorphims or piecewise linear homeomorphisms is trivial if $r<n-1$. This confirms a conjecture on Zimmer's program for these manifolds.