Closure of Orbits of the Pure Mapping Class Group on the Character Variety

For every surface S, the pure mapping class group G_S acts on the (SL_2)-character variety Ch_S of a fundamental group P of S. The character variety Ch_S is a scheme over the ring of integers. Classically this action on the real points Ch_S(R) of the character variety has been studied in the context of the Teichmuller theory and SL(2,R)-representations of P. In a seminal work, Goldman studied this action on a subset of Ch_S(R) which comes from SU(2)-representations of P. In this case, Goldman showed that if S is of genus g>1 and zero punctures, then the action of G_S is ergodic. Previte and Xia studied this question from topological point of view, and when g>0, proved that the orbit closure is as large as algebraically possible. Bourgain, Gamburd, and Sarnak studied this action on the F_p-points Ch_S(F_p) of the character variety where S is a puncture torus. They conjectured that in this case, this action has only two orbits, where one of the orbits has only one point. Recently, this conjecture was proved for large enough primes by Chen. When S is an n-punture sphere, the finite orbits of this action on Ch_S(C) are connected to the algebraic solutions of Painleve differential equations. In this talk, I will report on our recent contributions to this theory. Here are some sample results:

An almost complete description of the Zariski-closure of infinite G_S-orbits in Ch_S(F) where F is a characteristic zero field. Answering a question of Goldman-Previte-Xia by understanding the orbit closure of G_S on SU(2)-representation part of Ch_S(R) where S is an n-puncture sphere. Show that the original result of Previte and Xia is not accurate and give a description of the cases where it fails. Proving that in most cases the closure of G_S-orbits in the p-adic integer points Ch_S(Z_p) are open within given polynomial constrains. We give precise description of exceptional cases. (This is a joint work with Natallie Tamam.)