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Patterson-Sullivan measures for transverse subgroups

Richard Canary, Tengren Zhang, Andrew Zimmer

Abstract

We study Patterson-Sullivan measures for a class of discrete subgroups of higher rank semisimple Lie groups, called transverse groups, whose limit set is well-defined and transverse in a partial flag variety. This class of groups includes both Anosov and relatively Anosov groups, as well as all discrete subgroups of rank one Lie groups. We prove an analogue of the Hopf-Tsuji-Sullivan dichotomy and then use this dichotomy to prove a variant of Burger's Manhattan curve theorem. We also use the Patterson-Sullivan measures to obtain conditions for when a subgroup has critical exponent strictly less than the original transverse group. These gap results are new even for Anosov groups.

Patterson-Sullivan measures for transverse subgroups

Abstract

We study Patterson-Sullivan measures for a class of discrete subgroups of higher rank semisimple Lie groups, called transverse groups, whose limit set is well-defined and transverse in a partial flag variety. This class of groups includes both Anosov and relatively Anosov groups, as well as all discrete subgroups of rank one Lie groups. We prove an analogue of the Hopf-Tsuji-Sullivan dichotomy and then use this dichotomy to prove a variant of Burger's Manhattan curve theorem. We also use the Patterson-Sullivan measures to obtain conditions for when a subgroup has critical exponent strictly less than the original transverse group. These gap results are new even for Anosov groups.
Paper Structure (37 sections, 55 theorems, 298 equations)

This paper contains 37 sections, 55 theorems, 298 equations.

Table of Contents

  1. Introduction
  2. The geometric framework for the proofs
  3. Background and notation
  4. Semisimple Lie groups
  5. The Weyl group and the opposition involution
  6. Parabolic subgroups and flag manifolds
  7. Cartan projection
  8. Weights and partial Cartan projections
  9. The partial Iwasawa cocycle
  10. When $\theta$ is symmetric
  11. Discrete subgroups of semisimple Lie groups
  12. Critical exponents
  13. $\mathop{\mathrm{\mathsf{P}}}\nolimits_\theta$-divergent groups
  14. $\mathop{\mathrm{\mathsf{P}}}\nolimits_\theta$-transverse groups
  15. Anosov groups
  16. ...and 22 more sections

Key Result

Theorem 1.2

Suppose $\Gamma\subset\mathsf{PSL}(d,\mathbb K)$ is a non-elementary $\mathop{\mathrm{\mathsf{P}}}\nolimits_\theta$-transverse subgroup, $\phi\in\mathfrak{a}_\theta^*$ and $\delta^\phi(\Gamma) < +\infty$. If $G$ is a subgroup of $\Gamma$ such that $Q_G^\phi(\delta^\phi(G))=+\infty$ and $\Lambda_\the

Theorems & Definitions (93)

  • Definition 1.1
  • Theorem 1.2: see Theorem \ref{['entropy drop']}
  • Corollary 1.3: see Corollary \ref{['entropy drop anosov']}
  • Theorem 1.4: see Proposition \ref{['prop: consequences of shadow lemma']}, Proposition \ref{['prop: support on conical limit set']}, Corollary \ref{['cor: ergodicity']} and Corollary \ref{['cor:uniqueness']}
  • Theorem 1.5: see Theorem \ref{['manhattan curve']}
  • Corollary 1.6: see Corollary \ref{['cor:manhattan curve Z dense case']}
  • Proposition 1.7: see Proposition \ref{['prop:shadow estimates']}
  • Proposition 2.3
  • Lemma 2.4: Quint quint-ps
  • Lemma 2.5: Quint quint-ps
  • ...and 83 more