Bi-Lipschitz rigidity of discrete subgroups

TL;DR

This paper proves a bi-Lipschitz rigidity theorem for Zariski dense discrete subgroups in simple real algebraic groups and shows such groups cannot have smooth slim limit sets in certain homogeneous spaces.

Contribution

It introduces a new bi-Lipschitz rigidity result for Zariski dense discrete subgroups and applies it to limit set smoothness in higher rank semisimple groups.

Findings

Zariski dense discrete subgroups exhibit bi-Lipschitz rigidity

Such subgroups cannot have $C^1$-smooth slim limit sets in $G/P$

The results apply to higher rank semisimple algebraic groups

Abstract

We obtain a bi-Lipschitz rigidity theorem for a Zariski dense discrete subgroup of a connected simple real algebraic group. As an application, we show that any Zariski dense discrete subgroup of a higher rank semisimple algebraic group $G$ cannot have a $C^1$-smooth slim limit set in $G/P$ for any non-maximal parabolic subgroup $P$.