Simone Coccia

Density of integral points on character varieties

Given a smooth quasi-projective complex variety Y with a simple normal crossings compactification, a (relative) SL_2-character variety of Y is a moduli space parametrizing SL_2-representations of the fundamental group of Y with fixed traces along the boundary components of the compactification. Well-known examples of SL_2-character varieties are Markoff-type cubic surfaces, and in recent years the study of their integral points has attracted much attention, notably with the work of Bourgain, Gamburd and Sarnak. In this talk I will present joint work with Daniel Litt where we prove that integral points are potentially Zariski dense in every SL_2-character variety (provided the fixed traces along the boundary components are algebraic integers). The proof uses work of Corlette and Simpson to reduce to the case of Y a Riemann surface, where we produce an integral point whose orbit under the mapping class group action is Zariski dense.