Constructing thin subgroups of $\mathrm{SL}(n+1,\mathbb{R})$ via bending

Abstract

We use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite-volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in $SO\left(n,1\right)$ it is possible to find infinitely many noncommensurable lattices in $SL\left(n+1,ℝ\right)$ that contain a thin subgroup isomorphic to a finite-index subgroup of the original arithmetic lattice. This class of arithmetic lattices includes all noncocompact arithmetic lattices as well as all cocompact arithmetic lattices when $n$ is even.