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 $G\subset \mathrm{Isom}{H}_{\mathbb{F}}^n$. Introducing a nilpotent Sierpiński 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 $G\subset \mathrm{Isom}{H}_{\mathbb{F}}^n$ 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.

14 pages, 1 figure