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Sublinearly Morseness in Higher Rank Symmetric Spaces

Rou Wen

TL;DR

This work develops a sublinear Morse theory in higher rank symmetric spaces by extending Kapovich–Leeb–Porti notions to $P_{ heta}$-sublinearly Morse sequences. It builds a framework using Patterson–Sullivan measures on partial flag manifolds, convex projective models, and Bowen–Margulis–Sullivan flow to show that, for non-elementary, $P_{ heta}$-transverse and $oldsymbol{ u}$-divergent groups with finite BM–S measure, the sublinearly Morse boundary has full Patterson–Sullivan measure. A key contribution is the equivalence between sublinearly Morse sequences and convergence of partial flag data, and a sublinearly Morse Lemma connecting these sequences to Weyl-cone geometry via asymptotic cones. The results yield a robust description of hyperbolic-like behavior in higher rank spaces, with corollaries for relatively Anosov representations and a clear geometric interpretation in terms of diamonds, Weyl cones, and parallel sets. Overall, this work generalizes 1D hyperbolic Morse phenomena to a sublinear, higher-rank setting, providing tools for boundary measure rigidity and geometric analysis of discrete subgroups.

Abstract

The goal of this paper is to develop a theory of "sublinearly Morse boundary" and prove a corresponding sublinearly Morse lemma in higher rank symmetric space of non-compact type. This is motivated by the work of Kapovich-Leeb-Porti and the theory of sublinearly Morse quasi-geodesics developed in the context of CAT(0) geometry.

Sublinearly Morseness in Higher Rank Symmetric Spaces

TL;DR

This work develops a sublinear Morse theory in higher rank symmetric spaces by extending Kapovich–Leeb–Porti notions to -sublinearly Morse sequences. It builds a framework using Patterson–Sullivan measures on partial flag manifolds, convex projective models, and Bowen–Margulis–Sullivan flow to show that, for non-elementary, -transverse and -divergent groups with finite BM–S measure, the sublinearly Morse boundary has full Patterson–Sullivan measure. A key contribution is the equivalence between sublinearly Morse sequences and convergence of partial flag data, and a sublinearly Morse Lemma connecting these sequences to Weyl-cone geometry via asymptotic cones. The results yield a robust description of hyperbolic-like behavior in higher rank spaces, with corollaries for relatively Anosov representations and a clear geometric interpretation in terms of diamonds, Weyl cones, and parallel sets. Overall, this work generalizes 1D hyperbolic Morse phenomena to a sublinear, higher-rank setting, providing tools for boundary measure rigidity and geometric analysis of discrete subgroups.

Abstract

The goal of this paper is to develop a theory of "sublinearly Morse boundary" and prove a corresponding sublinearly Morse lemma in higher rank symmetric space of non-compact type. This is motivated by the work of Kapovich-Leeb-Porti and the theory of sublinearly Morse quasi-geodesics developed in the context of CAT(0) geometry.

Paper Structure

This paper contains 20 sections, 28 theorems, 167 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Settings and Preliminaries
  3. Lie group theory
  4. Parabolic Subgroup and Flag Manifold
  5. Patterson--Sullivan Measures
  6. Properly Convex Domain
  7. Sublinear Rays
  8. Transverse Representations
  9. Bowen--Margulis--Sullivan Measure
  10. Hopf--Tsuji--Sullivan Dichotomy
  11. The Morse Boundary
  12. The sublinearly Morse Boundary
  13. Main Theorem
  14. Equivalent Condition of sublinearly Morseness
  15. Proof of Main Theorem
  16. ...and 5 more sections

Key Result

Theorem 1

For a $P_{\theta}$-transverse subgroup $\Gamma$ satisfying certain conditions, there exists a Patterson--Sullivan measure $\mu$ associated to $\Gamma$. If $\Gamma$ is not $P_{\theta}$-Anosov, then the set of endpoints of the undistorted $\theta$-uniform regular sequences in $\Gamma$ has measure zero

Figures (1)

  • Figure :

Theorems & Definitions (69)

  • Definition 1.1
  • Theorem : Theorem \ref{['zero measure in non Anosov']}
  • Definition 1.2: $P_{\theta}$-Sublinearly Morse sequence
  • Theorem : Corollary \ref{['sublin. Morse lemma for seq']}
  • Corollary : Theorem \ref{['genericity in rel Anosov']}
  • Remark 2.3
  • Definition 2.4: Parabolic Subgroup
  • Remark 2.5
  • Definition 2.6: $P_{\theta}$-divergent
  • Definition 2.7: $P_{\theta}$-transverse
  • ...and 59 more