Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rigidity of locally symmetric rank one manifolds of infinite volume

Boris N. Apanasov

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 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 $G\subset\operatorname{Isom}H_\mathbb{F}^n$ whose non-trivial deformations are induced by $G$-equivariant homeomorphisms of the space. Here we consider two situations: either the limit set $Λ(G)$ is the whole sphere at infinity $\partial H_\mathbb{F}^n$ or restrictions of such non-trivial deformations to components of the discontinuity set $Ω(G)\subset \partial H_\mathbb{F}^n$ 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.

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 of symmetric rank one space over division algebra compatible with the action of a discrete group . Introducing a nilpotent Sierpiński carpet with a positive Lebesgue measure in the nilpotent geometry in and its stretching, we construct a non-rigid discrete -hyperbolic groups whose non-trivial deformations are induced by -equivariant homeomorphisms of the space. Here we consider two situations: either the limit set is the whole sphere at infinity or restrictions of such non-trivial deformations to components of the discontinuity set are given by restrictions of -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.
Paper Structure (8 sections, 3 theorems, 12 equations)

This paper contains 8 sections, 3 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric spaces of rank one
  4. Upper half-space model of $H_{{\mathbb F}}^ n$
  5. Spinal spheres in $\partial H_{{\mathbb F}}^ n$, bisectors and inversions
  6. Spherical CR-structures
  7. Discrete groups with $\Lambda(G)=\partial H_{{\mathbb F}}^ n$
  8. Deformations of ${\mathbb F}$-hyperbolic manifolds trivial in the boundary at infinity

Key Result

Theorem 1.1

Let $\Gamma_1, \Gamma_2\subset\operatorname{Isom}H_{{\mathbb F}}^ m$ be isomorphic lattices in ${\mathbb F}$-hyperbolic space $H_{{\mathbb F}}^ m$ with real dimension $n\geq 3$, that is isomorphic discrete isometry groups with finite volume quotients $H_{{\mathbb F}}^ m/\Gamma_i<\infty$ over real ${

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 3.1
  • Remark 3.2
  • Remark 4.1
  • Remark 4.2