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Connectedness of the Gromov boundary of fine curve graphs

Yusen Long, Dong Tan

TL;DR

The work analyzes the Gromov boundary of the fine curve graph $\\mathcal{C}^f(S)$ for high-genus surfaces, proving a bounded geodesic image theorem and deducing the boundary is not compact. By passing to a carefully chosen subgraph $\\mathcal{NC}^f_{o,\\pitchfork}(S)$, the authors adapt Wright-type connectedness criteria to show the boundary is path-connected, and under a visual metric with a non-separating base curve, linearly connected. The paper also establishes that the boundary action of $\\mathrm{Homeo}(S)$ is minimal and of general type, with the limit set equal to the entire boundary, while highlighting the close relationship between the fine and surviving curve graphs through quasi-isometries. Overall, the results advance understanding of the large-scale geometry and boundary topology of the fine curve graph, with implications for the dynamics of surface homeomorphisms and bordifications.

Abstract

The fine curve graph was introduced to study homeomorphism group of surfaces. In this paper we study the topology of the Gromov boundary of this graph for closed surfaces with higher genus. We first prove a bounded geodesic image theorem for the fine curve graph, a consequence of which is the non-compactness of the Gromov boundary. Using this theorem, we are able to show that the Gromov boundary is linearly connected with respect to some visual metric.

Connectedness of the Gromov boundary of fine curve graphs

TL;DR

The work analyzes the Gromov boundary of the fine curve graph for high-genus surfaces, proving a bounded geodesic image theorem and deducing the boundary is not compact. By passing to a carefully chosen subgraph , the authors adapt Wright-type connectedness criteria to show the boundary is path-connected, and under a visual metric with a non-separating base curve, linearly connected. The paper also establishes that the boundary action of is minimal and of general type, with the limit set equal to the entire boundary, while highlighting the close relationship between the fine and surviving curve graphs through quasi-isometries. Overall, the results advance understanding of the large-scale geometry and boundary topology of the fine curve graph, with implications for the dynamics of surface homeomorphisms and bordifications.

Abstract

The fine curve graph was introduced to study homeomorphism group of surfaces. In this paper we study the topology of the Gromov boundary of this graph for closed surfaces with higher genus. We first prove a bounded geodesic image theorem for the fine curve graph, a consequence of which is the non-compactness of the Gromov boundary. Using this theorem, we are able to show that the Gromov boundary is linearly connected with respect to some visual metric.
Paper Structure (18 sections, 43 theorems, 50 equations, 3 figures)

This paper contains 18 sections, 43 theorems, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Gromov hyperbolic space and bordification
  3. Definition.
  4. Visual metric and boundary connectedness.
  5. Fine curve graph and its subgraphs
  6. Fine curve graph.
  7. Distance estimation.
  8. Non-separating subgraph.
  9. Bounded geodesic image theorem
  10. Subsurface projection.
  11. Surgery.
  12. Proof of bounded geodesic image theorem.
  13. A surviving version.
  14. Application: non-compactness.
  15. Connectivity properties
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $S$ be an orientable connected closed surface with genus $g\geq 3$, then the Gromov boundary $\partial\mathcal{C}^\dagger(S)$ of the fine curve graph is path connected. Moreover, if $o$ is a non-separating curve on $S$, then equipped with the visual metric $\rho_{o,b}$, the Gromov boundary is li

Figures (3)

  • Figure 1: A possible subsurface projection $\pi_Y(\alpha)$ of curve $\alpha$ on subsurface $Y$.
  • Figure 2: Surgery for \ref{['K1']} (left) and \ref{['K3']} (right).
  • Figure 3: Surgery for \ref{['K2']} and its proof of Lemma \ref{['lem_surgery_surviving']}.

Theorems & Definitions (90)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Bounded image theorem
  • Theorem 1.4
  • Definition 2.1: Gromov hyperbolic space
  • Definition 2.2: Gromov boundary
  • Proposition 2.3
  • proof : Sketch of Proof
  • Remark 2.4
  • Theorem 2.5: Väsäilä
  • ...and 80 more