A survey of topological Zimmer's program
英文摘要:
In this article, we survey the status of topological Zimmer's conjecture on matrix group actions on manifolds.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
Shengkui Ye
In this article, we survey the status of topological Zimmer's conjecture on matrix group actions on manifolds.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
继续探索
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.
We obtain a sufficient and necessary condition for a finite group to act effectively on a closed flat manifold. Let \ $G=E_{n}(R)$, $EU_{n}(R,Λ),$ $\mathrm{SAut}(F_{n})$ 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.
We consider Zimmer's program of lattice actions on surfaces by PL homomorphisms. It is proved that when the surface is not the torus or Klein bottle the action of any finite-index subgroup of SL(n,Z), n>4, (more generally for any 2-big lattice), factors through a finite group action. The proof is based on an establishment of a PL version of Reeb-Thurston's stability.
This paper can be viewed as a sequel to the author's long survey on the Zimmer program \cite{F11} published in 2011. The sequel focuses on recent rapid progress on certain aspects of the program particularly concerning rigidity of Anosov actions and Zimmer's conjecture that there are no actions in low dimensions. Some emphasis is put on the surprising connections between these two different sets of developments and also on the key connections and ideas for future research that arise from these works taken together.
Let $\mathrm{SL}_{n}(\mathbb{Z})$ $(n\geq 3)$ be the special linear group and $M^{r}$ be a closed aspherical manifold. It is proved that when $r<n,$ a group action of $\mathrm{SL}_{n}(\mathbb{Z})$ on $M^{r}$ by homeomorphisms is trivial if and only if the induced group homomorphism $\mathrm{SL}_{n}(% \mathbb{Z})\rightarrow \mathrm{Out}(π_{1}(M))$ is trivial. For (almost) flat manifolds, we prove a similar result in terms of holonomy groups. Especially, when $π_{1}(M)$ is nilpotent, the group $\mathrm{SL}_{n}(% \mathbb{Z})$ cannot act nontrivially on $M$ when $r<n.$ This confirms a conjecture r…
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