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Patterson-Sullivan measures for relatively Anosov groups

Richard Canary, Andrew Zimmer, Tengren Zhang

TL;DR

The paper develops a comprehensive Patterson–Sullivan theory for relatively Anosov subgroups of semisimple Lie groups, proving that the $\phi$-Poincaré series diverges at its finite critical exponent and deriving existence, uniqueness and ergodicity results for $\phi$-Patterson–Sullivan measures. It then obtains an entropy gap for peripheral subgroups and a strict concavity statement for entropies, by combining divergence results with a resolution-of-singularities analysis of peripheral growth and multiplicative Cartan-estimates. The approach relies on reductions to no-simple-factor linear models, the Groves–Manning cusp space, and a detailed analysis of the partial Cartan and Iwasawa structures, linking higher-rank dynamics to classical rank-one intuition. Collectively, these results extend Patterson–Sullivan theory to higher-rank, relatively Anosov contexts, with implications for convergence dichotomies, limit cones (Benoist cones), and the geometry of relatively quasiconvex subgroups.

Abstract

We establish existence, uniqueness and ergodicity results for Patterson-Sullivan measures for relatively Anosov groups. As applications we obtain an entropy gap theorem and a strict concavity result for entropies associated to linear functionals.

Patterson-Sullivan measures for relatively Anosov groups

TL;DR

The paper develops a comprehensive Patterson–Sullivan theory for relatively Anosov subgroups of semisimple Lie groups, proving that the -Poincaré series diverges at its finite critical exponent and deriving existence, uniqueness and ergodicity results for -Patterson–Sullivan measures. It then obtains an entropy gap for peripheral subgroups and a strict concavity statement for entropies, by combining divergence results with a resolution-of-singularities analysis of peripheral growth and multiplicative Cartan-estimates. The approach relies on reductions to no-simple-factor linear models, the Groves–Manning cusp space, and a detailed analysis of the partial Cartan and Iwasawa structures, linking higher-rank dynamics to classical rank-one intuition. Collectively, these results extend Patterson–Sullivan theory to higher-rank, relatively Anosov contexts, with implications for convergence dichotomies, limit cones (Benoist cones), and the geometry of relatively quasiconvex subgroups.

Abstract

We establish existence, uniqueness and ergodicity results for Patterson-Sullivan measures for relatively Anosov groups. As applications we obtain an entropy gap theorem and a strict concavity result for entropies associated to linear functionals.
Paper Structure (27 sections, 50 theorems, 237 equations)

This paper contains 27 sections, 50 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Consequences of Theorem \ref{['intro: rel anosov divergent']}
  3. Outline of the proof of Theorem \ref{['intro: rel anosov divergent']}
  4. Conditions (1) and (2) are not equivalent to (3) in Theorem \ref{['thm:characterizing finite entropy functionals']}
  5. Semisimple Lie groups
  6. Cartan projection
  7. Parabolic subgroups and flag manifolds
  8. Fundamental weights and partial Cartan projections
  9. The partial Iwasawa cocycle
  10. The Linear Case
  11. Properties of unipotent subgroups
  12. Relatively hyperbolic groups
  13. Relatively hyperbolic groups
  14. The Groves--Manning cusp space
  15. Discrete subgroups of semisimple Lie groups
  16. ...and 12 more sections

Key Result

Theorem 1.1

If $\Gamma \subset \mathsf G$ is a $\mathop{\mathrm{\mathsf{P}}}\nolimits_\theta$-Anosov subgroup relative to $\mathcal{P}$, $\phi \in {\mathfrak a}_\theta^*$ and $\delta^\phi(\Gamma) < +\infty$, then $Q_\Gamma^\phi$ diverges at its critical exponent.

Theorems & Definitions (87)

  • Theorem 1.1: Theorem \ref{['rel anosov divergence']}
  • Remark 1.2
  • Theorem 1.3: see Section \ref{['sec:characterizing finite entropy functionals']}
  • Theorem 1.4: CZZ3
  • Corollary 1.5
  • Theorem 1.6: CZZ3
  • Corollary 1.7: Corollary \ref{['cor:entropy gap rel anosov 1']}
  • Theorem 1.8: CZZ3
  • Corollary 1.9
  • Corollary 1.10
  • ...and 77 more