Title: Hurwitz spaces, Nielsen equivalence, and Markoff triples
Abstract: In this talk we’ll describe a connection between three classes of problems, and how all three perspectives can be leveraged to answer some questions in number theory. We begin with the notion of “dessins d’enfants” (child’s drawings), or equivalently covers of the Riemann sphere branched above 3 points. The absolute Galois group of Q acts (faithfully) on the set of such objects with finite orbits. An old (and very open) question of Grothendieck asks for a classification of the orbits in terms of discrete invariants of covers. This question can be generalized to the problem of classifying connected components of Hurwitz spaces (moduli spaces of finite covers of curves of a fixed genus and fixed number of branch points). By covering space theory, the geometric version of the generalized problem is equivalent to the group theoretic problem of understanding mapping class group orbits on the set of surjections from a surface group P to a finite group G. Such problems arose in the 1950’s out of a desire to understand group presentations, and recently has enjoyed renewed interest due to its relevance to the product replacement algorithm. If G is a finite group of Lie type, such surjections can then be understood as finite-field valued points in a certain character variety (a moduli space of representations of P). If P is a free group of rank 2, and G is the group SL(2,p), then under these correspondences, the problem of understanding connected components of the relevant Hurwitz space corresponds to the question of whether the Markoff equation x^2 + y^2 + z^2 - xyz = 0 satisfies “strong approximation” at p (i.e., do its integral points surject onto its mod p points?). We will explain how arithmetic results of Bourgain, Gamburd, and Sarnak on the character variety side can be combined with geometric results on the Hurwitz space side to establish strong approximation for the Markoff equation. As a corollary, we answer a question of Frobenius from 1913.