Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Birman-Hilden theory for big mapping class groups

Nestor Colin, Ruben Hidalgo, Rita Jiménez Rolland, Israel Morales, Saúl Quispe

TL;DR

This work extends Birman–Hilden theory to big mapping class groups by proving that fully ramified branched covers $p:S\to X$ between surfaces with $S$ of infinite type or $X$ of negative Euler characteristic satisfy the Birman–Hilden property. The authors combine the Alexander method with Klein-surface techniques to treat orientable and non-orientable cases in a unified framework, enabling infinite-sheeted and infinite-type scenarios. They derive injective realizations of non-orientable mapping class groups into orientable doubles and establish injections for surface braid groups under orientable doubling, thereby generalizing several finite-type results to the infinite-type setting. The results provide structural insights into big mapping class groups and offer systematic constructions of geometrically characteristic covers with robust lift properties.

Abstract

Let $S$ and $X$ be two connected topological surfaces without boundary, and assume that $S$ is either of infinite type or has negative Euler characteristic. In this paper, we prove that if $p:S\rightarrow X$ is a fully ramified branched covering map, then $p$ satisfies the Birman-Hilden property. This generalizes a theorem of Winarski, and the known results in the literature, to the context of surfaces of infinite type and branched covering maps of infinite degree. As an application, we show that the mapping class group (respectively, the braid group on $k$-strands) of a non-orientable surface of infinite type can be realized as a subgroup of the mapping class group (respectively, the braid group on $2k$-strands) of its orientable double cover.

Birman-Hilden theory for big mapping class groups

TL;DR

This work extends Birman–Hilden theory to big mapping class groups by proving that fully ramified branched covers between surfaces with of infinite type or of negative Euler characteristic satisfy the Birman–Hilden property. The authors combine the Alexander method with Klein-surface techniques to treat orientable and non-orientable cases in a unified framework, enabling infinite-sheeted and infinite-type scenarios. They derive injective realizations of non-orientable mapping class groups into orientable doubles and establish injections for surface braid groups under orientable doubling, thereby generalizing several finite-type results to the infinite-type setting. The results provide structural insights into big mapping class groups and offer systematic constructions of geometrically characteristic covers with robust lift properties.

Abstract

Let and be two connected topological surfaces without boundary, and assume that is either of infinite type or has negative Euler characteristic. In this paper, we prove that if is a fully ramified branched covering map, then satisfies the Birman-Hilden property. This generalizes a theorem of Winarski, and the known results in the literature, to the context of surfaces of infinite type and branched covering maps of infinite degree. As an application, we show that the mapping class group (respectively, the braid group on -strands) of a non-orientable surface of infinite type can be realized as a subgroup of the mapping class group (respectively, the braid group on -strands) of its orientable double cover.
Paper Structure (16 sections, 17 theorems, 20 equations)

This paper contains 16 sections, 17 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Relation with known results in the literature
  3. Some applications to geometrically characteristic covers
  4. Preliminaries
  5. Mapping class group of surfaces
  6. The Alexander method
  7. Birman-Hilden theory for branched covers
  8. Klein surfaces and auxiliary results
  9. Klein surfaces and some double covers
  10. Klein surfaces and branched covers
  11. Proof of Theorem \ref{['BirmanHildenProperty-Generalization']}
  12. An alternative proof for unbranched regular covers
  13. Some applications to geometrically characteristic covers
  14. Injections between mapping class groups
  15. Mapping class group of non-orientable surfaces
  16. ...and 1 more sections

Key Result

Theorem 1

Let $S$ and $X$ be two connected topological surfaces without boundary, where $S$ is either of infinite type or of finite type with negative Euler characteristic. If $p:S\to X$ is a fully ramified branched covering map, then $p$ has the Birman-Hilden property.

Theorems & Definitions (41)

  • Theorem 1: Birman--Hilden property for big mapping class groups
  • Corollary 2
  • Remark 1.1
  • Corollary 3: Injections between mapping class groups
  • Corollary 4: Mapping class groups as quotients
  • Corollary 5: Mapping class groups and braid groups of a non-orientable surface and its orientable double cover
  • Definition 2.1: Alexander system
  • Definition 2.2: k-stable Alexander system
  • Theorem 2.3: Alexander's method for surfaces
  • Definition 2.4: Branched covering map
  • ...and 31 more