Hilbert’s Irreducibility Theorem via Random Walks

Abstract

Abstract Let $G$ be a connected linear algebraic group over a number field $K$, let $\Gamma $ be a finitely generated Zariski dense subgroup of $G(K)$, and let $Z\subseteq G(K)$ be a thin set, in the sense of Serre. We prove that, if $G/\textrm {R}_{u}(G)$ is either trivial or semisimple and $Z$ satisfies certain necessary conditions, then a long random walk on a Cayley graph of $\Gamma $ hits elements of $Z$ with negligible probability. We deduce corollaries to Galois covers, characteristic polynomials, and fixed points in group actions. We also prove analogous results in the case where $K$ is a global function field.