Block mapping class groups and their finiteness properties

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 ${\mathcal{C}}_d$.

v2: Fixes an error in Proposition 6.6 of v1, main results unaffected. Streamlined exposition. 19 pages, 1 figure