Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program

TL;DR

This paper proves that certain real-analytic actions of SL(n,Z) on the n-torus are conjugate to standard linear actions outside a finite set of periodic orbits, under specific measure-preserving conditions.

Contribution

It establishes a rigidity result for real-analytic SL(n,Z) actions on tori, confirming a case of Zimmer's program with measure and homotopy constraints.

Findings

Actions are conjugate to linear models outside periodic orbits

Rigidity holds for ergodic measures with support not contained in a ball

Results apply for n ≥ 3 with standard homotopy data

Abstract

We prove that any real-analytic action of $SL(n,\Z), n\ge 3$ with standard homotopy data that preserves an ergodic measure $\mu$ whose support is not contained in a ball, is analytically conjugate on an open invariant set to the standard linear action on the complement to a finite union of periodic orbits.