Equidistribution of hyperbolic groups in homogeneous spaces (Q6624845)

A fundamental question in ergodic theory is to understand what kind of averaging operators satisfy ergodic theorems for measure-preserving group actions. For actions of amenable groups, \textit{E. Lindenstrauss} [Invent. Math. 146, No. 2, 259--295 (2001; Zbl 1038.37004)]\N showed that averaging along Følner sets gives versions of the standard Birkhoff ergodic theorem under relatively mild conditions. For the action of hyperbolic groups, the story is more complicated. \textit{A. Nevo} and \textit{E. M. Stein} [Acta Math. 173, No. 1, 135--154 (1994; Zbl 0837.22003)]\Nproved an ergodic theorem for the action of free groups, and generalizations to different settings were obtained by, among others, \textit{K. Fujiwara} and \textit{A. Nevo} [Ergodic Theory Dyn. Syst. 18, No. 4, 843--858 (1998; Zbl 0919.22002)], \textit{A. Bufetov} [Transl., Ser. 2, Am. Math. Soc. 202, 39--50 (2001; Zbl 1002.47004); Ann. Math. (2) 155, No. 3, 929--944 (2002; Zbl 1028.37001)], \textit{L. Bowen} [Ergodic Theory Dyn. Syst. 30, No. 1, 97--129 (2010; Zbl 1205.37007)], and \textit{A. I. Bufetov} and \textit{C. Series} [Math. Proc. Camb. Philos. Soc. 151, No. 1, 145--159 (2011; Zbl 1219.22008)]. In all of these cases, the averaging is done by taking the Cesaro average of averages over spheres in the Cayley graph with respect to a finite generating set. In all of these (and some newer results as well) the results give convergence of averages for almost every basepoint. \N\NThe paper under review gives very precise results which hold for \textit{every} point, provided the space being acted up on is a homogeneous space \(G/\Lambda\), and the acting group \(\Gamma\) is a hyperbolic group acting on \(G/\Lambda\) via a representation \(\rho: \Gamma \rightarrow G\) with \(\mbox{Ad}\)-Zariski dense image. The authors show that unless \(\Gamma\), the orbit of \(x\), is finite, the Cesaro averages of sphere averages of the \(\Gamma\)-action equidistribute with respect to Haar measure. They can obtain even stronger results for actions on the flat torus, where they can drop Cesaro averaging. Key ideas are the use of results of \textit{Y. Benoist} and \textit{J.-F. Quint} [Ann. Math. (2) 174, No. 2, 1111--1162 (2011; Zbl 1241.22007); Jpn. J. Math. (3) 7, No. 2, 135--166 (2012; Zbl 1268.22010); Ann. Math. (2) 178, No. 3, 1017--1059 (2013; Zbl 1279.22013); J. Am. Math. Soc. 26, No. 3, 659--734 (2013; Zbl 1268.22011)], and the theory of geodesic combings.