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Automorphisms of fine curve graphs for nonorientable surfaces

Mitsuaki Kimura, Erika Kuno

TL;DR

The paper extends the seminal result on automorphisms of the fine curve graph from orientable to closed nonorientable surfaces of genus $g\ge 4$, showing $\mathrm{Aut}(\mathcal{C}^{\dagger}(N))\cong \mathrm{Homeo}(N)$. It adapts the Long–Margalit–Pham–Verberne–Yao framework to handle one-sided curves by introducing the extended graph $\mathcal{EC}^{\dagger}(N)$ and proving that automorphisms preserve two-sided structures such as torus/pants/bigon configurations and related hull/link properties. A key step is constructing an extension map from $\mathrm{Aut}(\mathcal{C}^{\dagger}(N))$ to $\mathrm{Aut}(\mathcal{EC}^{\dagger}(N))$ and then recovering a homeomorphism from an automorphism of the extended graph via limit curves, establishing surjectivity. The approach also proves connectedness of relevant fine arc graphs in the nonorientable setting, paralleling Ivanov-type results and strengthening the correspondence between topological symmetries and combinatorial automorphisms for nonorientable surfaces.

Abstract

The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb as a graph consisting of essential simple closed curves on the surface. Long, Margalit, Pham, Verberne, and Yao proved that the automorphism group of the fine curve graph of a closed orientable surface is isomorphic to the homeomorphism group of the surface. In this paper, based on their argument, we prove that the automorphism group of the fine curve graph of a closed nonorientable surface $N$ of genus $g \geq 4$ is isomorphic to the homeomorphism group of $N$.

Automorphisms of fine curve graphs for nonorientable surfaces

TL;DR

The paper extends the seminal result on automorphisms of the fine curve graph from orientable to closed nonorientable surfaces of genus , showing . It adapts the Long–Margalit–Pham–Verberne–Yao framework to handle one-sided curves by introducing the extended graph and proving that automorphisms preserve two-sided structures such as torus/pants/bigon configurations and related hull/link properties. A key step is constructing an extension map from to and then recovering a homeomorphism from an automorphism of the extended graph via limit curves, establishing surjectivity. The approach also proves connectedness of relevant fine arc graphs in the nonorientable setting, paralleling Ivanov-type results and strengthening the correspondence between topological symmetries and combinatorial automorphisms for nonorientable surfaces.

Abstract

The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb as a graph consisting of essential simple closed curves on the surface. Long, Margalit, Pham, Verberne, and Yao proved that the automorphism group of the fine curve graph of a closed orientable surface is isomorphic to the homeomorphism group of the surface. In this paper, based on their argument, we prove that the automorphism group of the fine curve graph of a closed nonorientable surface of genus is isomorphic to the homeomorphism group of .
Paper Structure (9 sections, 22 theorems, 3 equations, 1 figure)

This paper contains 9 sections, 22 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonorientable surfaces
  4. Torus pairs, pants pairs and bigon pairs
  5. Graph theory
  6. Key propositions
  7. Connectedness of fine arc graphs
  8. Automorphisms of the extended fine curve graphs
  9. Automorphisms of the fine curve graph

Key Result

Theorem 1.1

For $g\geq 4$, the natural map $\eta\colon\mathrm{Homeo}(N_{g})\rightarrow\mathrm{Aut}(\mathcal{C}^{\dagger}(N_{g}))$ is an isomorphism.

Figures (1)

  • Figure 1: Bigon pair for disjoint inessential curves bounding a disk in the nested case (left) and the unnested case (right).

Theorems & Definitions (32)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1: cf. Long--Margalit--Pham--Verberne--Yao
  • Proposition 3.2: cf. Long--Margalit--Pham--Verberne--Yao
  • Lemma 3.3: cf. Long--Margalit--Pham--Verberne--Yao
  • Lemma 3.4: cf. Long--Margalit--Pham--Verberne--Yao
  • proof : Proof of Lemma \ref{['presv_hulls']}
  • Remark 3.5
  • Lemma 3.6
  • ...and 22 more