Zariski-dense surface groups in non-uniform lattices of split real Lie groups

TL;DR

This paper constructs thin Hitchin representations in non-uniform lattices of certain split real Lie groups using arithmetic methods, demonstrating their abundance and the existence of infinitely many mapping class group orbits.

Contribution

It provides the first explicit construction of thin Hitchin representations in non-uniform lattices of split real Lie groups, expanding understanding of their geometric and arithmetic properties.

Findings

Existence of thin Hitchin representations in non-uniform lattices of specified Lie groups.

Infinitely many mapping class group orbits for these representations.

Every non-uniform lattice in SL(p,R) with prime p ≠ 2 contains such representations.

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.