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Exhaustion of $\mathcal{C}(N)$ via rigid expansions

Jesús Hernández Hernández, Cristhian E. Hidber

Abstract

Let $N$ be a connected closed non-orientable surface of genus at least 6. In this work we prove that there exists a finite subgraph $\mathfrak{X}$ (Irmak's finite rigid set from ``Elmas Irmak. Exhausting curve complexes by finite rigid sets on nonorientable surfaces. J.Topol.Anal., 16(2):261--289, 2024'') such that any graph endomorphism $\varphi$ of $\mathcal{C}(N)$ whose restriction to $\mathfrak{X}$ is (locally) injective, $\varphi$ is induced by a homeomorphism of $N$. To prove this, we first prove that $\mathfrak{X}$ and its rigid expansions exhaust $\mathcal{C}(N)$.

Exhaustion of $\mathcal{C}(N)$ via rigid expansions

Abstract

Let be a connected closed non-orientable surface of genus at least 6. In this work we prove that there exists a finite subgraph (Irmak's finite rigid set from ``Elmas Irmak. Exhausting curve complexes by finite rigid sets on nonorientable surfaces. J.Topol.Anal., 16(2):261--289, 2024'') such that any graph endomorphism of whose restriction to is (locally) injective, is induced by a homeomorphism of . To prove this, we first prove that and its rigid expansions exhaust .
Paper Structure (21 sections, 32 theorems, 138 equations, 26 figures, 3 tables)

This paper contains 21 sections, 32 theorems, 138 equations, 26 figures, 3 tables.

Table of Contents

  1. Preliminaries
  2. Rigid expansions
  3. Curve graph
  4. Model for $N_{g}$ and the subgraph $\mathfrak{Y} \leq \mathcal{C}(N)$
  5. Useful additional curves
  6. The curves $\varepsilon_{i,j}$ and the set $\mathcal{E}$
  7. The curves $\alpha_{i}$
  8. The curves $\mu^{1}_{i}$
  9. The curves $\alpha_{i,j}$
  10. The curves $\nu^{\pm}_{i,j}$
  11. Translations of $\mathfrak{Y}$ using generators
  12. Szepietowski generating set
  13. $\tau_{\gamma_{i}}^{\pm 1}(\mathfrak{Y}) \subset \mathfrak{Y}^{11}$
  14. Proof of Lemma \ref{['lemma:dtgammaiA']}
  15. Proof of Lemma \ref{['lemma:dtgammaiM']}
  16. ...and 6 more sections

Key Result

Theorem 1

Let $N$ be a connected closed non-orientable surface of genus $g \geq 6$, and let $\varphi: \mathcal{C}(N) \to \mathcal{C}(N)$ be a graph morphism. If $\varphi|_{\mathfrak{X}}$ is (locally) injective, then $\varphi$ is induced by a homeomorphism

Figures (26)

  • Figure 3: The curves $\varepsilon_{2,5}$ in red and $\varepsilon_{6,2}$ in blue.
  • Figure 4: The curve $\varepsilon_{1,4}$ in purple.
  • Figure 5: The curve $\alpha_{1}$ in red.
  • Figure 6: The curves of the set $A_{1}$ in green, those of the set $A_{2}$ in red, those of the set $E_{1}$ in blue, and those of the set $E_{2}$ in purple. These curves uniquely determine the curve $\mu^{1}_{2}$.
  • Figure 7: The curves of the set $A$ in red, those of the set $E_{1}$ in blue, those of the set $E_{2}$ in purple, those of the set $E_{3}$ in orange, and the curve of the set $M$ in green. These curves uniquely determine the curve $\alpha_{2,5}$ in fuchsia.
  • ...and 21 more figures

Theorems & Definitions (54)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary : C,JHH2
  • Lemma 1.1
  • proof
  • Corollary 1.2
  • Lemma 1.3
  • Theorem 1.4
  • Lemma 1.5
  • ...and 44 more