Title: Symmetries of flat manifolds, Jordan property and the general Zimmer program,
Abstract: We obtain a sufficient and necessary condition for a finite group that could
act effectively on closed flat manifolds. Let $G=E_{n}(R)$ the elementary
subgroup of a linear group, $EU_{n}(R,\Lambda )$ the elementary subgroup of a
unitary group, $\mathrm{SAut}(F_{n})$ the special automorphism group of a free
group or $\mathrm{SOut}(F_{n})$ the special outer automorphism group of a free
group. 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.