Zariski dense surface groups in \(\operatorname{SL}(2k + 1, \mathbb{Z})\) (Q6541438)

Let \(G\) be a linear algebraic group over \(\mathbb Q\) and \(\Gamma\) an arithmetic subgroup. An infinite index subgroup \(\Lambda\subset\Gamma\) is called a \textit{thin group}, if it is Zariski dense in \(G\). Thin groups appear in various contexts and surprisingly share many properties with arithmetic groups. See [\textit{P. Sarnak}, Math. Sci. Res. Inst. Publ. 61, 343--362 (2014; Zbl 1365.11039)] for more on this.\N\NIf \(G\) is simple, it is hard to construct thin groups, which are not essentially free groups. The present paper gives a construction of thin groups in \(\mathrm{SL}_n(\mathbb{Z})\) for odd \(n\), which are isomorphic to fundamental groups of compact Riemann surfaces, aka surface groups. The construction uses a so-called bending process, which works for surface groups only, as it uses geometry of the Riemann surface. One basically starts with a representation and then conjugates a geometrically defined part of the representation. Repeated application of bending yields a group with the desired properties.