Integral Zariski dense surface groups in $\operatorname{SL}(n,\mathbf{R})$

TL;DR

This paper demonstrates that certain integral representations of surface groups over a number field can be deformed into Zariski dense representations in SL(n,R), extending previous methods to higher dimensions.

Contribution

It generalizes a known deformation technique to produce Zariski dense surface groups in SL(n,R) from integral representations over number fields.

Findings

Existence of Zariski dense surface groups in SL(n,R)

Deformation of integral representations preserving properties

Extension of previous methods to higher dimensions

Abstract

Given a number field $K$, we show that certain $K$-integral representations of closed surface groups can be deformed to being Zariski dense while preserving many useful properties of the original representation. This generalizes a method due to Long and Thistlethwaite who used it to show that thin surface groups in $\operatorname{SL}(2k+1,\mathbf{Z})$ exist for all $k$.