Counting Representations of Fuchsian Groups Over Finite Fields

Recent progress on character bounds for groups of Lie type makes it feasible in many cases to find the asymptotic growth, for fixed q and n tending to infinity, of the number of n-dimensional representations of a Fuchsian group G over the field with q elements. Applications include determining the dimensions of complex representation varieties of Fuchsian groups in high degree and proving the existence of strongly Zariski-dense Fuchsian subgroups, in the sense of Breuillard-Green-Guralnick-Tao, of classical groups of high rank.