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

Abstract

We obtain a sufficient and necessary condition for a finite group to act effectively on a closed flat manifold. Let \ $G=E_n(R)$, $E{U}_n(R,\Lambda ),$ $\mathrm{SAut}(F_n)$ or $\mathrm{SOut}(F_n).$ As applications, we prove that when $n\ge 3$ 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.

Some English words are improved. No math changes, accepted by the Journal of the London Mathematical Society

References in corpus (4)

• Diffeomorphism Groups of Compact 4-manifolds are not always Jordan
• Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds
• Euler characteristics and actions of automorphism groups of free groups
• The action of matrix groups on aspherical manifolds

Cited by in corpus (2)

• Property FW, differentiable structures, and smoothability of singular actions
• Actions of large finite groups on aspherical manifolds