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Non-planar ends are continuously unforgettable

Javier Aramayona, Rodrigo De Pool, Rachel Skipper, Jing Tao, Nicholas G. Vlamis, Xiaolei Wu

TL;DR

The paper addresses rigidity of big mapping class groups for infinite-genus, non-planar-end surfaces by proving that any continuous epimorphism between suitably large subgroups is induced by a surface homeomorphism. The authors develop a generating set for compactly supported mapping classes via an Alexander chain, analyze Dehn twists and multitwists, and leverage lantern and braid-type relations to show twists map to twists in irreducible, continuous epimorphisms, enabling a passage to exhaustions and a global homeomorphism. They extend the main result from the compactly supported subgroup to larger large subgroups, including perfectly self-similar manifolds, and deduce Hopfian properties for these big mapping class groups. Finally, they relax the continuity hypothesis to a purely algebraic condition—infinitely multiplicative on twists—preserving the geometric conclusion and broadening the scope of the rigidity phenomenon in big mapping class groups.

Abstract

We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups.

Non-planar ends are continuously unforgettable

TL;DR

The paper addresses rigidity of big mapping class groups for infinite-genus, non-planar-end surfaces by proving that any continuous epimorphism between suitably large subgroups is induced by a surface homeomorphism. The authors develop a generating set for compactly supported mapping classes via an Alexander chain, analyze Dehn twists and multitwists, and leverage lantern and braid-type relations to show twists map to twists in irreducible, continuous epimorphisms, enabling a passage to exhaustions and a global homeomorphism. They extend the main result from the compactly supported subgroup to larger large subgroups, including perfectly self-similar manifolds, and deduce Hopfian properties for these big mapping class groups. Finally, they relax the continuity hypothesis to a purely algebraic condition—infinitely multiplicative on twists—preserving the geometric conclusion and broadening the scope of the rigidity phenomenon in big mapping class groups.

Abstract

We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups.
Paper Structure (10 sections, 32 theorems, 14 equations, 4 figures)

This paper contains 10 sections, 32 theorems, 14 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Surfaces and curves
  4. Mapping class groups and their topology
  5. Twists
  6. Generating compactly supported mapping classes
  7. Irreducible epimorphisms
  8. Proof of the main theorem
  9. Moving to the full mapping class group
  10. Relaxing continuity

Key Result

Corollary 1.3

If $M$ is an orientable infinite-genus 2-manifold with no planar ends, then every mostly pure large subgroup of $\mathop{\mathrm{MCG}}\nolimits(M)$ is a Hopfian topological group; in particular, $\mathop{\mathrm{PMCG}}\nolimits(M)$ is a Hopfian topological group, and if $M$ has finitely many ends, t

Figures (4)

  • Figure 1: The Dehn twists about the curves shown generate $\mathop{\mathrm{PMCG}}\nolimits(S)$, where $S$ is the surface shown. Here, $S$ is a genus three surface with three boundary components.
  • Figure 2: The Alexander chain constructed in Theorem \ref{['thm:generators']} on a 2-ended infinite-genus 2-manifold.
  • Figure 3: The surfaces $F_2$ (left) and $F_3$ (right) and their corresponding Alexander chains, $C_2$ and $C_3$.
  • Figure 4: The curves involved in the inductive step in the proof of Theorem \ref{['thm:generators']}.

Theorems & Definitions (54)

  • Definition 1.1
  • Corollary 1.3
  • Corollary 1.5
  • Corollary 1.6
  • Proposition 2.1
  • Lemma 2.2: Primer
  • Lemma 2.3: Primer
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 44 more