Block mapping class groups and their finiteness properties

TL;DR

This paper introduces block mapping class groups for Cantor surfaces, analyzing their finiteness properties and establishing connections with symmetric Thompson groups, revealing new subgroups with specific finiteness types.

Contribution

It defines block mapping class groups with prescribed local actions and proves their finiteness properties relate directly to the local subgroup H, extending known asymptotic mapping class groups.

Findings

When blocks are spheres or tori, the groups are of type F_n if and only if H is of type F_n.

Existence of subgroups of Map(C_d) with specific finiteness properties containing all compact subsurface mapping class groups.

Provides a positive answer to a question about the relation between block mapping class groups and symmetric Thompson groups.

Abstract

A Cantor surface $\mathcal C_d$ is a non-compact surface obtained by gluing copies of a fixed compact surface $Y^d$ (a block), with $d+1$ boundary components, in a tree-like fashion. For a fixed subgroup $H<Map(Y^d)$ , we consider the subgroup $\mathfrak B_d(H)<Map(\mathcal C_d)$ whose elements eventually send blocks to blocks and act like an element of $H$; we refer to $\mathfrak B_d(H)$ as the block mapping class group with local action prescribed by $H$. The family of groups so obtained contains the asymptotic mapping class groups of \cite{SW21a,ABF+21, FK04}. Moreover, there is a natural surjection onto the family symmetric Thompson groups of Farley--Hughes \cite{FH15}; in particular, they provide a positive answer to \cite[Question 5.37]{AV20}. We prove that, when the block is a (holed) sphere or a (holed) torus, $\mathfrak B_d(H)$ is of type $F_n$ if and only if $H$ is of type $F_n$. As a consequence, for every $n$, $Map(C_d)$ has a subgroup of type $F_n$ but not $F_{n+1}$ which contains the mapping class group of every compact subsurface of $\mathcal C_d$.