An extension criterion for lattice actions on the circle (Q2908706)

Let \(\Gamma\) be a lattice in a locally compact second countable group \(G\). The aim of this paper is to give a necessary and sufficient condition for a \(\Gamma\)-action by homeomorphisms of the circle to extend continuously to \(G\). This condition is expressed in terms of the real bounded Euler class of this action. Combined with classical vanishing theorems in bounded cohomology, the author recovers some known rigidity results in a unified manner.NEWLINENEWLINEThe main result of this paper is:NEWLINENEWLINETheorem 1.2. Let \(\Gamma \subset G\) be a lattice in a locally compact, second countable group \(G\) and \(\rho : \Gamma \to \text{Homeo}^+ (S^1)\) be a minimal unbounded action. Then the following conditions are equivalent:NEWLINENEWLINE1) The bounded real Euler class \(\rho^\ast (e^b_{\mathbb{R}})\) of \(\rho \) is in the image of the restriction map \(H^2_{bc} (G, \mathbb{R}) \to H^2_b (\Gamma,\mathbb{R})\) .NEWLINENEWLINE2) The strongly proximal factor \(\rho_{sp}\) of \(\rho\) extends continuously to \(G\).NEWLINENEWLINEFor the entire collection see [Zbl 1225.00042].