Zariski dense surface subgroups in $SL(n,\mathbb{Q})$ with odd $n$

TL;DR

This paper constructs a path of Zariski dense, discrete, and faithful representations of a surface group into SL(n,R) for odd n, with all representations rational and lying in SL(n,Q).

Contribution

It provides the first explicit construction of such rational, Zariski dense surface subgroups in SL(n,Q) for odd n.

Findings

Existence of a path of rational, Zariski dense surface subgroups in SL(n,Q)

Construction of discrete, faithful representations for odd n

Representations are contained in SL(n,Q) for all rational parameters

Abstract

For odd $n$ we construct a path $\rho_t\colon \pi_1(S) \to SL(n,\mathbb{R})$ of discrete, faithful and Zariski dense representations of a surface group such that $\rho_t(\pi_1(S)) \subset SL(n,\mathbb{Q})$ for every $t\in \mathbb{Q}$.