# Block mapping class groups and their finiteness properties

## TL;DR

This paper studies mapping class groups of surfaces and constructs subgroups with specific finiteness properties.

## Contribution

The paper introduces a new family of subgroups with controllable finiteness properties and answers an open question about Thompson groups.

## Key findings

- A family of subgroups is constructed that surjects onto symmetric Thompson groups.
- The finiteness properties of the subgroups match those of a prescribed subgroup H.
- Subgroups of type F_n but not F_{n+1} are constructed for mapping class groups of surfaces.

## Abstract

Given \documentclass[12pt]{minimal}
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\begin{document}$$g \in \mathbb N \cup \{0, \infty \}$$\end{document}g∈N∪{0,∞}, let \documentclass[12pt]{minimal}
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\begin{document}$$\Sigma _g$$\end{document}Σg denote the closed surface of genus g with a Cantor set removed, if \documentclass[12pt]{minimal}
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\begin{document}$$g<\infty $$\end{document}g<∞; or the blooming Cantor tree, when \documentclass[12pt]{minimal}
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\begin{document}$$g= \infty $$\end{document}g=∞. We construct a family \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak B(H)$$\end{document}B(H) of subgroups of \documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{Map}\,}}(\Sigma _g)$$\end{document}Map(Σg) whose elements preserve a block decomposition of \documentclass[12pt]{minimal}
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\begin{document}$$\Sigma _g$$\end{document}Σg, and eventually like act like an element of H, where H is a prescribed subgroup of the mapping class group of the block. The group \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak B(H)$$\end{document}B(H) surjects onto an appropriate symmetric Thompson group of Farley–Hughes; in particular, it answers positively. Our main result asserts that \documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak B(H)$$\end{document}B(H) is of type \documentclass[12pt]{minimal}
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\begin{document}$$F_n$$\end{document}Fn if and only if H is. As a consequence, for every \documentclass[12pt]{minimal}
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\begin{document}$$g\in \mathbb N \cup \{0, \infty \}$$\end{document}g∈N∪{0,∞} and every \documentclass[12pt]{minimal}
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\begin{document}$$n\ge 1$$\end{document}n≥1, we construct a subgroup \documentclass[12pt]{minimal}
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\begin{document}$$G <{{\,\textrm{Map}\,}}(\Sigma _g)$$\end{document}G<Map(Σg) that is of type \documentclass[12pt]{minimal}
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\begin{document}$$F_n$$\end{document}Fn but not of type \documentclass[12pt]{minimal}
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\begin{document}$$F_{n+1}$$\end{document}Fn+1, and which contains the mapping class group of every compact surface of genus \documentclass[12pt]{minimal}
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\begin{document}$$\le g$$\end{document}≤g and with non-empty boundary.