Can lattices in \(\mathrm{SL}(n,\mathbb R)\) act on the circle? (Q2908711)

The author discusses some aspects concerning the actions of large groups on small manifolds. First, he presents a proof of the following theorem: Let \(\Gamma \) be a finite-index subgroup of \(\mathrm{SL}(n, \mathbb{R})\) with \(n\geq 3\). Then \(\Gamma\) has no faithful action on the circle \(S^1\). The proof is obtained by exploiting the interaction between some nilpotent subgroups of \(\Gamma\). The author presents some aspects concerning the following conjecture related to the considered arguments: Let \(\Gamma\) be a lattice in \(\mathrm{SL}(n, \mathbb{R})\), with \(n\geq 3\). Then \(\Gamma\) has no faithful action on \(S^1\). Then the author presents the Reeb-Thurston stability theorem with a proof in a special case and an outline of proof in the case where \(\Lambda\) has the Kazhdan property (T). Next he studies smooth actions of Kazhdan groups on the circle and Ghys's proof that actions have a finite orbit. Some auxiliary results concerning the amenability and ergodicity theorem are presented. At the end, he discusses the bounded cohomology and the Burger-Monod proof.NEWLINENEWLINEFor the entire collection see [Zbl 1225.00042].