Rigidity of locally symmetric rank one manifolds of infinite volume

Abstract

We discuss questions by Mostow \cite{Mo1}, Bers \cite{B} and Krushkal \cite{Kr1, Kr2} about uniqueness of a conformal or spherical CR structure on the sphere at infinity $\partial H_\mathbb{F}^n$ of symmetric rank one space $H_\mathbb{F}^n$ over division algebra $\mathbb{F}=\mathbb{R}\,,\mathbb{C}\,,\mathbb{H}\,,\text{or}\,\, \mathbb{O} $ compatible with the action of a discrete group $G\subset\operatorname{Isom}H_\mathbb{F}^n$. Introducing a nilpotent Sierpi\'{n}ski carpet with a positive Lebesgue measure in the nilpotent geometry in $\partial H_\mathbb{F}^n\setminus\{\infty\}$ and its stretching, we construct a non-rigid discrete $\mathbb{F}$-hyperbolic groups $G\subset\operatorname{Isom}H_\mathbb{F}^n$ whose non-trivial deformations are induced by $G$-equivariant homeomorphisms of the space. Here we consider two situations: either the limit set $\Lambda(G)$ is the whole sphere at infinity $\partial H_\mathbb{F}^n$ or restrictions of such non-trivial deformations to components of the discontinuity set $\Omega(G)\subset \partial H_\mathbb{F}^n$ are given by restrictions of $\mathbb{F}$-hyperbolic isometries. In both cases the demonstrated non-rigidity is related to non-ergodic dynamics of the discrete group action on the limit set which could be the whole sphere at infinity.