# Symmetries of flat manifolds, Jordan property and the general Zimmer program

## TL;DR

This paper characterizes when finite groups act effectively on flat manifolds, proving triviality of certain actions for specific groups and confirming a conjecture related to Zimmer's program, also establishing Jordan properties for homeomorphism groups.

## Contribution

It provides a necessary and sufficient condition for effective finite group actions on flat manifolds and confirms a conjecture in Zimmer's program for these manifolds.

## Key findings

- Finite groups like $E_{n}(R)$ act trivially on certain flat manifolds.
- Homeomorphism groups of flat manifolds are Jordan with dimension-dependent constants.
- Confirmed conjecture related to Zimmer's program for flat manifolds.

## Abstract

We obtain a sufficient and necessary condition for a finite group to act effectively on a closed flat manifold. Let \ $G=E_{n}(R)$, $EU_{n}(R,\Lambda ),$ $\mathrm{SAut}(F_{n})$ or $\mathrm{SOut}(F_{n}).$ As applications, we prove that when $n\geq 3$ every group action of $G$ on a closed flat manifold $M^{k}$ ($k<n$) by homeomorphisms is trivial. This confirms a conjecture related to Zimmer's program for flat manifolds. Moreover, it is also proved that the group of homeomorphisms of closed flat manifolds are Jordan with Jordan constants depending only on dimensions.