Zariski-dense Hitchin representations in uniform lattices

TL;DR

This paper constructs Zariski-dense surface subgroups within uniform lattices of certain split real Lie groups using Hitchin representations, demonstrating their abundance and distribution across various lattices.

Contribution

It establishes the existence of infinitely many Zariski-dense Hitchin representations in uniform lattices of several split real Lie groups, expanding understanding of their subgroup structures.

Findings

Existence of Zariski-dense surface subgroups in multiple Lie groups.

Infinitely many mapping class group orbits of Hitchin representations.

All lattices of Sp(4,R) contain a Zariski-dense surface subgroup.

Abstract

We construct Zariski-dense surface subgroups in infinitely many commensurability classes of uniform lattices of the split real Lie groups $\operatorname{SL}(n,\mathbb{R})$, $\operatorname{Sp}(2n,\mathbb{R})$, $\operatorname{SO}(k+1,k)$, and $\operatorname{G}_2$. These subgroups are images of Hitchin representations. In particular, we show that every uniform lattice of $\operatorname{Sp}(2n,\mathbb{R})$, of $\operatorname{SO}(k+1,k)$ with $k\equiv1,2[4]$ and of $\operatorname{G}_2$ contains infinitely many mapping class group orbits of Zariski-dense Hitchin representations of fixed genus. Together with Long-Thistlethwaite and with a previous paper of the author, it implies that all lattices of $\operatorname{Sp}(4,\mathbb{R})$ contain a Zariski-dense surface subgroup.