Zariski-dense surface groups in non-uniform lattices of split real Lie groups
Jacques Audibert
TL;DR
The paper constructs thin Hitchin surface subgroups inside non-uniform lattices of split real Lie groups by combining arithmetic and geometric techniques. It develops a framework using $ ext{R}/ ext{Q}$-forms of split groups, controlled by non-abelian Galois cohomology and compatible cocycles, to locate $ au_n$-images inside explicit $ ext{Q}$-arithmetic subgroups such as $ ext{SL}(n, ext{Z})$, $ ext{SU}( ext{I}_n,\sigma; ext{Z}[ oot 0pt ext√{d}])$, and related unitary forms; it then performs bending along carefully chosen curves to produce Zariski-dense Hitchin representations. The key technical contributions include a detailed classification of $ ext{Q}$-arithmetic subgroups containing $ au_n( ext{Γ})$, a construction of explicit thin Hitchin representations in several split groups (including $ ext{SL}(n)$, $ ext{Sp}(2n)$, $ ext{SO}(k+1,k)$, and $ extbf{G}_2$), and a bend-based strategy together with strong approximation arguments to produce infinitely many $ ext{MCG}(S_g)$-orbits of thin representations. These results imply that every non-uniform lattice in many split groups contains thin Hitchin representations, significantly advancing the understanding of thin subgroups in higher-rank arithmetic lattices and their mapping class group dynamics.
Abstract
For $\textrm{SL}(n,\mathbb{R})$ ($n\geq3$), $\textrm{SO}(n+1,n)$ ($n\geq2$), $\textrm{Sp}(2n,\mathbb{R})$ ($n\geq2$) and for the adjoint real split form of the exceptional group $\textrm{G}_2$, we exhibit non-uniform lattices in which we construct thin Hitchin representations by arithmetic methods. These representations give infinitely many orbits under the action of the mapping class group (except maybe for $\textrm{G}_2$). In particular, we show that when $p\neq2$ is prime every non-uniform lattice of $\mathrm{SL}(p,\mathbb{R})$ contains thin Hitchin representations.