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Classification of horospherical invariant measures in higher rank

Inhyeok Choi, Dongryul M. Kim

Abstract

Let $G$ be a product of rank-one simple real algebraic groups and let $Γ< G$ be a Zariski dense Anosov subgroup, or relatively Anosov subgroup. In this paper, we prove a complete classification of invariant Radon measures for the maximal horospherical action on $Γ\backslash G$. In particular, when $Γ$ is Anosov, this solves the open problems proposed by Landesberg--Lee--Lindenstrauss--Oh for $\operatorname{rank} G \le 3$, and by Oh in general. More generally, we consider the horospherical foliation of a product of $\operatorname{CAT}(-1)$ spaces, and present a classification of Radon measures supported on a recurrent subfoliation that are invariant under the action of transverse subgroups.

Classification of horospherical invariant measures in higher rank

Abstract

Let be a product of rank-one simple real algebraic groups and let be a Zariski dense Anosov subgroup, or relatively Anosov subgroup. In this paper, we prove a complete classification of invariant Radon measures for the maximal horospherical action on . In particular, when is Anosov, this solves the open problems proposed by Landesberg--Lee--Lindenstrauss--Oh for , and by Oh in general. More generally, we consider the horospherical foliation of a product of spaces, and present a classification of Radon measures supported on a recurrent subfoliation that are invariant under the action of transverse subgroups.

Paper Structure

This paper contains 34 sections, 39 theorems, 173 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results for Anosov subgroups
  3. Beyond Anosov subgroups
  4. On the proof
  5. Organization
  6. Acknowledgements
  7. Notation
  8. Basic CAT(--1) geometry
  9. Contracting property and squeezing property
  10. Alignment
  11. Shadows and alignment
  12. Isometries
  13. Non-elementary subgroups of isometries
  14. Horofunctions
  15. Conical limit sets
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\Gamma <G$ be a Zariski dense Anosov subgroup and $v \in \operatorname{int} \mathfrak{a}^+$. Let $\mathcal{L}_{\Gamma} \subset \mathfrak{a}^+$ denote the limit coneThe limit cone of $\Gamma$ is the asymptotic cone of the Cartan projections of $\Gamma$ in $\mathfrak{a}$. We will revisit this lat

Figures (2)

  • Figure 1: A squeezing geodesic $\gamma$
  • Figure 2: Alignment of geodesics and points.

Theorems & Definitions (80)

  • Theorem 1.1: LLLO_Horospherical
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Remark 1.11
  • Remark 1.11
  • ...and 70 more