中科院数学与系统科学研究院

数学研究所

学术报告

几何拓扑研讨班

报告人：Peter Smillie (Max Planck Institut für Mathematik, Leipzig）

题  目：The mapping class group action on SL(2,R) representations

时  间：2024.9.10（周二）下午4:00-5:00

地  点：晨兴 410

摘  要：Let X(G) be the character variety of representations of a surface group into a Lie group G, so that the mapping class group acts on X(G). When G is compact, this action is ergodic by work of Goldman and Pickrell-Xia, and a well-known conjecture of Goldman is that for G=PSL(2,R) (and genus at least 3), the action is ergodic on each non-Fuchsian topological component of X(G). This turns out to be essentially equivalent to a conjecture of Bowditch that every non-elementary non-Fuchsian representation in X(PSL(2,R)) sends some simple closed curve loop to a non-hyperbolic element. By studying the action of the mapping class group on the tangent cone to the subvariety of nontrivial diagonal representations, we prove that Bowditch's condition holds in a neighborhood of the nontrivial diagonal representations in the Euler number zero component. This is joint work with James Farre and Martin Bobb.