A recording of a talk at Big Ideas in Dynamics in Geometry: A Hybrid Conference. The talk took place on September 28th, 2024 at UIC.

Abstract: In this talk, given a (hyperbolizable) surface S (with boundary) we will describe the action of the mapping class group Mod(S) on the (relative) SL(2, C)–character variety X(S) := Hom(π1(S), SL(2, C))/SL(2, C), and define an open domain of discontinuity of the action which (in many cases) strictly contains the interior of the set of discrete and faithful representations.

In the first part of the talk, we will focus on the original idea of Bowditch and consider S=S_{1,1} the once-holed torus. We will describe the combinatorial methods suggested by Bowditch based on the relationship between points in the SL(2, C)–character variety X(S) and Markoff triples, the main steps of the proofs and its generalization by Tan-Wong-Zhang.

In the second part of the talk, we will focus on joint work with Palesi, Tan, and Lawton, and generalize the description to various cases, considering S = S_{0,4} the four-holed sphere, S= N_{1,3} the three-holed projective plane and replacing SL(2,C) with SU(2,1). For each of these generalizations, we will explain how to modify the set-up discussed in the first part of the talk, and the difficulties that arise when one tries to do a general discussion. Time-permitting, we will also discuss relations with the work of Minsky on primitive-stable representations and Schlich.