Multitwists in big mapping class groups

TL;DR

The paper proves that for infinite-type surfaces, the subgroup generated by multitwists cannot generate the closure of the compactly supported mapping class group, since it fails to contain the Torelli group. The authors construct an explicit element in $\mathcal{I}(S)$ as an infinite product of high-translation-length partial pseudo-Anosov maps on disjoint finite-type subsurfaces and certify its exclusion from the multitwist subgroup using a Bestvina–Bromberg–Fujiwara stability framework. The core technique couples nondisplaceable finite-type subsurfaces with projection-geometry and quasimorphism arguments to force a contradiction with any finite multitwist expression. The Loch Ness monster case is addressed via Birman exact sequences, ensuring the result holds uniformly across infinite-type surfaces.

Abstract

We show that the closure of the compactly supported mapping class group of an infinite-type surface is not generated by the collection of multitwists (i.e. products of powers of twists about disjoint non-accumulating curves).

Figures (1)

• Figure 1: The subsurfaces of Lemma \ref{['lem:nondisp']}

Theorems & Definitions (9)

• Theorem 1
• Lemma 1
• proof
• Proposition 2
• Lemma 3
• proof
• Proposition 4
• proof
• proof : Proof of Theorem \ref{['thm:main']}