Superstrong approximation for monodromy groups

Abstract: This document is an expanded version of a lecture presented at a conference on "Thin Groups and Superstrong Approximation" held at the Mathematical Sciences Research Institute in February 2012. Superstrong approximation is a criterion on a finitely generated group, saying that certain Cayley graphs associated to finite quotients of the group form an expander family. In recent years, our knowledge about superstrong approximation for infinite-index Zariski-dense subgroups of arithmetic lattices ("thin groups") has drastically improved. We briefly survey the construction of monodromy groups, discuss our (limited) knowledge about whether such groups are thin, and discuss an application to arithmetic geometry (see the paper "Expander graphs, gonality, and variation of Galois representations") deriving from recent advances in superstrong approximation. We conclude by indulging in some speculations about more general contexts, asking: what are the interesting questions about "nonabelian superstrong approximation" and "superstrong approximation for Galois groups?" We discuss the relation of these notions with the Product Replacement Algorithm and the Bogomolov property for infinite algebraic extensions of number fields.