Block mapping class groups and their finiteness properties.

Abstract

Given $g\in N\cup \{0,\infty \}$ , let ${\Sigma }_g$ denote the closed surface of genus g with a Cantor set removed, if $g<\infty$ ; or the blooming Cantor tree, when $g=\infty$ . We construct a family $B\left(H\right)$ of subgroups of $\text{Map}\left({\Sigma }_g\right)$ whose elements preserve a block decomposition of ${\Sigma }_g$ , 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 $B\left(H\right)$ surjects onto an appropriate symmetric Thompson group of Farley-Hughes; in particular, it answers positively. Our main result asserts that $B\left(H\right)$ is of type $F_n$ if and only if H is. As a consequence, for every $g\in N\cup \{0,\infty \}$ and every $n\ge 1$ , we construct a subgroup $G<\text{Map}\left({\Sigma }_g\right)$ that is of type $F_n$ but not of type $F_{n+1}$ , and which contains the mapping class group of every compact surface of genus $\le g$ and with non-empty boundary.