Inhyeok Choi, Invariant measures for subgroups of mapping class groups and Anosov groups скачать в хорошем качестве

Inhyeok Choi, Invariant measures for subgroups of mapping class groups and Anosov groups

Let S be a finite-type hyperbolic surface. Classical work by Thurston, Masur and Veech implies that the space ML(S) of measured laminations on S admits an essentially unique MCG(S)-invariant measure in the Lebesgue class. Building on this, Hamenstadt and Lindenstrauss—Mirzakhani independently classified all locally finite MCG(S)-invariant measures on ML(S). Their main ingredients include mixing of the Teichmuller flow, non-divergence of the unipotent flow and/or the train track theory for measured laminations. In this talk, I will explain how this result generalizes to non-elementary subgroups G of mapping class groups. Specifically, we construct a locally finite and G-invariant measure on ML(S) and prove that any ergodic invariant Radon measure on the G-recurrent measured laminations coincides with this measure. Our key ingredient is a geometric group theoretical property of pseudo-Anosov mapping classes, namely, the contracting property and the squeezing property. If time allows, I will sketch an analogous result about horospherical-invariant measures for Anosov groups. This talk is based on joint work with Dongryul M. Kim.