Arithmetic representations of fundamental groups, II: Finiteness

Let X be a smooth curve over finitely generated field k, and let l prime different from the characteristic of k. We analyze dynamics Galois action on deformation rings mod representations geometric fundamental group X. Using this analysis, we prove several finiteness results for function fields algebraically closed in arbitrary characteristic, weak variant Frey–Mazur conjecture 0. For example, show that if is normal, connected variety C, then (typically infinite) set π1(Xan) into GLn(Ql‾), which come geometry, has no limit points. As corollary, deduce L finite extension Ql, GLn(L), arise finite.