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The joint translation spectrum and Manhattan manifolds

Stephen Cantrell, Eduardo Reyes, Cagri Sert

Abstract

We define and study geometric versions of the Benoist limit cone and matrix joint spectrum, which we call the translation cone and the joint translation spectrum, respectively. These new notions allow us to generalize the study of embeddings into products of rank-one simple Lie groups and to compare group actions on different metric spaces, quasi-morphisms, Anosov representations and many other natural objects of study. We identify the joint translation spectrum with the image of the gradient function of a corresponding Manhattan manifold: a higher dimensional version of the well known and studied Manhattan curve. As a consequence we deduce many properties of the spectrum. For example we show that it is given by the closure of the set of all possible drift vectors associated to finitely supported, symmetric, admissible random walks on the associated group.

The joint translation spectrum and Manhattan manifolds

Abstract

We define and study geometric versions of the Benoist limit cone and matrix joint spectrum, which we call the translation cone and the joint translation spectrum, respectively. These new notions allow us to generalize the study of embeddings into products of rank-one simple Lie groups and to compare group actions on different metric spaces, quasi-morphisms, Anosov representations and many other natural objects of study. We identify the joint translation spectrum with the image of the gradient function of a corresponding Manhattan manifold: a higher dimensional version of the well known and studied Manhattan curve. As a consequence we deduce many properties of the spectrum. For example we show that it is given by the closure of the set of all possible drift vectors associated to finitely supported, symmetric, admissible random walks on the associated group.

Paper Structure

This paper contains 36 sections, 53 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Hyperbolic metric potentials
  3. Translation cone and joint translation spectrum
  4. Random walk and dynamical spectra
  5. Manhattan manifolds
  6. Manhattan manifolds as metric structures and its boundary
  7. Examples
  8. Preliminaries
  9. Metric notions
  10. Hyperbolic groups and metric structures
  11. The Gromov boundary and tempered potentials
  12. Geodesic currents
  13. The Mineyev flow
  14. Hyperbolic metric potentials
  15. Structural properties
  16. ...and 21 more sections

Key Result

Theorem 1.3

Let $\mathbf{D}=(\psi_1,\ldots,\psi_n)$ be a finite tuple of elements in $\mathcal{H}_\Gamma$. Then, the translation cone $\mathcal{TC}(\mathbf{D})$ is convex and it coincides with the asymptotic cone of $\mathbf{D}(\Gamma)$. Moreover, it has non-empty interior if and only if $\psi_1,\ldots, \psi_n$

Theorems & Definitions (107)

  • Definition 1.1
  • Definition 1.2: Translation cone
  • Theorem 1.3: Translation cone
  • Theorem 1.4: Joint translation spectrum
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Definition 1.10
  • ...and 97 more