Block mapping class groups and their finiteness properties

Given $$ , let $$ denote the closed surface of genus g with a Cantor set removed, if $$ ; or the blooming Cantor tree, when $$ . We construct a family $$ of subgroups of $$ whose elements preserve a block decomposition of $$ , and eventually like act like an element of H, where H is a prescribed subgroup of the mapping class group of the block. The group $$ surjects onto an appropriate symmetric Thompson group of Farley-Hughes; in particular, it answers positively. Our main result asserts that $$ is of type $$ if and only if H is. As a consequence, for every $$ and every $$ , we construct a subgroup $$ that is of type $$ but not of type $$ , and which contains the mapping class group of every compact surface of genus $$ and with non-empty boundary.