Arithmetic representations and non-commutative Siegel linearization

I'll explain joint work with Borys Kadets, explaining how to prove the following theorem. Let $X$ be a curve over a finitely generated field $k$, and let $\ell$ be a prime different from the characteristic of $k$. Then there exists $N=N(X,\ell)$ such that any semisimple arithmetic representation of $\pi_1(X_{\bar k})$ into $GL_n(\overline{\mathbb{Z}_\ell})$, which is trivial mod $\ell^N$, is in fact trivial. This extends previous work of mine from characteristic zero to all characteristics. The main new idea is to introduce techniques from dynamics; in particular a non-commutative version of Siegel's linearization theorem. For example, this gives restrictions on the possible torsion subgroups of abelian varieties over function fields. I'll also explain some related joint work in progress with Eric Katz on $\ell$-adic analogues of non-Abelian Hodge theory.