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Uniqueness theorem for completely non-degenerate B-groups

A. A. Glutsyuk, Yu. S. Ilyashenko

TL;DR

The paper proves a uniqueness theorem for completely non-degenerate B-groups by showing that such a group is uniquely determined by its factor; specifically, two non-degenerate B-groups with conformally equivalent factors are Möbius conjugate. The authors develop a refined factor theory that uses an associated marked 2-complex $K$ and a homotopy class $[\varphi]$ of maps from the main surface $S_0$ to $K$, and they establish a Main Lemma guaranteeing the existence of a good representative $[\varphi]$ with lifting and stabilizer properties. The proof of the conjugacy theorem hinges on Marden’s extension results and Maskit’s descriptions of factors, and it proceeds by constructing a global conjugacy on the discontinuity set via a careful lifting argument and diagrammatic coherence. These results lay groundwork for a simultaneous uniformization framework for algebraic curves with variable topology and extend the understanding of factor data beyond quasi-Fuchsian groups.

Abstract

We prove that a completely non-degenerate B-group is uniquely determined by its factor: two such groups with conformally equivalent factors are Möbius conjugate. A similar property is inherent to the quasi-Fuchsian groups but not to degenerate B-groups. We also study the factor of a B-group as a triple: the main factor, the marked characteristic complex, and a homotopy class of maps of the first to the second one.

Uniqueness theorem for completely non-degenerate B-groups

TL;DR

The paper proves a uniqueness theorem for completely non-degenerate B-groups by showing that such a group is uniquely determined by its factor; specifically, two non-degenerate B-groups with conformally equivalent factors are Möbius conjugate. The authors develop a refined factor theory that uses an associated marked 2-complex and a homotopy class of maps from the main surface to , and they establish a Main Lemma guaranteeing the existence of a good representative with lifting and stabilizer properties. The proof of the conjugacy theorem hinges on Marden’s extension results and Maskit’s descriptions of factors, and it proceeds by constructing a global conjugacy on the discontinuity set via a careful lifting argument and diagrammatic coherence. These results lay groundwork for a simultaneous uniformization framework for algebraic curves with variable topology and extend the understanding of factor data beyond quasi-Fuchsian groups.

Abstract

We prove that a completely non-degenerate B-group is uniquely determined by its factor: two such groups with conformally equivalent factors are Möbius conjugate. A similar property is inherent to the quasi-Fuchsian groups but not to degenerate B-groups. We also study the factor of a B-group as a triple: the main factor, the marked characteristic complex, and a homotopy class of maps of the first to the second one.

Paper Structure

This paper contains 18 sections, 9 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. B-groups
  4. Associated $2$-complex and factors of $B$-groups
  5. Factors of quasi-Fuchsian groups and the uniqueness theorem for such groups
  6. Factors of non-degenerate B-groups and uniqueness theorem
  7. Marden's results and Conjugacy theorem
  8. The Main Lemma on non-degenerate $B$-groups
  9. Proof of Theorem \ref{['conjd']} modulo the Main Lemma
  10. Definition of the conjugacy in the main component
  11. Definition of the conjugacy in the complement $\Omega \setminus \Delta_0$
  12. Conjugacy relation. Proof of Theorem \ref{['conjd']}
  13. Proof of the Main Lemma
  14. Statement B
  15. Base of induction: $k = 0$.
  16. ...and 3 more sections

Key Result

Theorem 2.8

Let $G$ be a $B$-group with invariant component $\Delta_0$. Let $S_0 = \Delta_0 / G$ and let $K = K(S_0, \alpha _1, \dots , \alpha _k)$ be the associated $2$-complex. Then there is a marking $S_1^+, \dots , S^+_s$ on $K$, so that $\Omega (G)/ G = S_0 + S_1^+ + \dots + S^+_s$. Furthermore, if $\Delt

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 16 more