Finitely generated subgroups of arithmetic lattices are frequently encountered in geometry, where they appear as monodromy groups attached to the variation of cohomology in families of manifolds or algebraic varieties. Here we survey briefly the construction of monodromy groups, discuss our (limited) knowledge about whether such groups are arithmetic, and summarize the results of [Ellenberg et al. 2012], which derive an application to arithmetic geometry from recent advances in superstrong approximation [Helfgott 2008; Pyber and Szabó 2011; Breuillard et al. 2011; Salehi Golsefidy and Varjú 2012]. We conclude by indulging ourselves in some speculations about more general contexts, asking: what are the interesting questions about “nonabelian superstrong approximation” and “superstrong approximation for Galois groups”?

The simplest example of monodromy is provided by the topological notion of a covering space. Suppose given a path-connected base space B, endowed with a choice of basepoint b, and let ƒ : X → B be a covering space of degree n; that is, each point x ∊ X has a neighborhood U such that ƒ -1(U) is homeomorphic to n disjoint copies of U. In particular, ƒ -1(b) consists of n points. A paradigmatic example is provided by B = X = ₵ x, where the map ƒ is given by z ↦ zn, and b = 1.