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On Uniformly Perfect Morse Boundaries

Suzhen Han, Qing Liu

TL;DR

This work develops a robust framework for uniformly perfect Morse boundaries in proper geodesic metric spaces. It establishes that, when the Morse boundary $\partial_*X$ has at least three points, uniform perfectness is equivalent to Morse geodesic richness and center-exhaustiveness, with each condition implying Morse boundary rigidity. For finitely generated groups with nonempty Morse boundary, the Morse boundary is uniformly perfect and the group is Morse geodesically rich and boundary rigid, applying to broad classes such as acylindrically hyperbolic, Artin, and hierarchically hyperbolic groups. A central rigidity theorem is proved: for spaces with these properties, a boundary homeomorphism is induced by a quasi-isometry if and only if it satisfies natural geometric regularity conditions (bi-Hölder, quasi-conformal, quasi-symmetric, or $2$-stable and quasi-Möbius), and these notions are quasi-isometry invariants. The paper further shows how to extend boundary maps to quasi-isometries via a projection-based construction, establishing a concrete bridge between boundary geometry and large-scale geometry with potential implications for understanding group boundaries in geometric group theory.

Abstract

We introduce and geometrically characterize the notion of uniformly perfect Morse boundary for proper geodesic metric spaces. As a unifying result, we prove that the Morse boundary of any finitely generated, non-elementary group is uniformly perfect whenever it is nonempty. This theorem applies to a broad class of groups, including all acylindrically hyperbolic groups, Artin groups, and hierarchically hyperbolic groups. Furthermore, we establish a rigidity theorem for homeomorphisms between such boundaries: for any two spaces with uniformly perfect Morse boundaries, a homeomorphism is induced by a quasi-isometry if and only if it satisfies any one of several natural geometric conditions. These conditions include being bi-Hölder, quasi-conformal, quasi-symmetric, or $2$-stable and quasi-Möbius.

On Uniformly Perfect Morse Boundaries

TL;DR

This work develops a robust framework for uniformly perfect Morse boundaries in proper geodesic metric spaces. It establishes that, when the Morse boundary has at least three points, uniform perfectness is equivalent to Morse geodesic richness and center-exhaustiveness, with each condition implying Morse boundary rigidity. For finitely generated groups with nonempty Morse boundary, the Morse boundary is uniformly perfect and the group is Morse geodesically rich and boundary rigid, applying to broad classes such as acylindrically hyperbolic, Artin, and hierarchically hyperbolic groups. A central rigidity theorem is proved: for spaces with these properties, a boundary homeomorphism is induced by a quasi-isometry if and only if it satisfies natural geometric regularity conditions (bi-Hölder, quasi-conformal, quasi-symmetric, or -stable and quasi-Möbius), and these notions are quasi-isometry invariants. The paper further shows how to extend boundary maps to quasi-isometries via a projection-based construction, establishing a concrete bridge between boundary geometry and large-scale geometry with potential implications for understanding group boundaries in geometric group theory.

Abstract

We introduce and geometrically characterize the notion of uniformly perfect Morse boundary for proper geodesic metric spaces. As a unifying result, we prove that the Morse boundary of any finitely generated, non-elementary group is uniformly perfect whenever it is nonempty. This theorem applies to a broad class of groups, including all acylindrically hyperbolic groups, Artin groups, and hierarchically hyperbolic groups. Furthermore, we establish a rigidity theorem for homeomorphisms between such boundaries: for any two spaces with uniformly perfect Morse boundaries, a homeomorphism is induced by a quasi-isometry if and only if it satisfies any one of several natural geometric conditions. These conditions include being bi-Hölder, quasi-conformal, quasi-symmetric, or -stable and quasi-Möbius.
Paper Structure (19 sections, 27 theorems, 101 equations, 4 figures)

This paper contains 19 sections, 27 theorems, 101 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Boundaries of hyperbolic spaces
  4. The Morse boundary and Metric Morse boundary
  5. Uniform perfectness and Morse geodesic richness
  6. Center-exhaustive spaces
  7. Uniform Perfectness of Morse boundaries
  8. Morse boundary rigid spaces
  9. Morse geodesically rich spaces
  10. Proof of Theorem \ref{['thm:equivalence']}
  11. Morse geodesic richness and center-exhaustiveness
  12. Morse geodesic richness implies uniform perfectness
  13. Uniformly perfectness implies center-exhaustiveness
  14. Quasi-isometry invariants
  15. Establishment of Theorem \ref{['homeo']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $X$ be a proper geodesic metric space whose Morse boundary $\partial_*X$ contains at least $3$ points. Then the following statements are equivalent: Furthermore, each of the above equivalent conditions implies that $X$ is Morse boundary rigid.

Figures (4)

  • Figure 1: Caption
  • Figure 2: Caption
  • Figure 3: Caption
  • Figure 4: Picture of Case II in Theorem \ref{['MGR-UP']}

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Quasi-isometric Embedding and Quasi-isometry
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5: Slim triangles and Morse triangles
  • ...and 51 more