Given \(g \in \mathbb N \cup \{0, \infty \}\) , 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 \(\mathfrak B(H)\) of subgroups of \({{\,\textrm{Map}\,}}(\Sigma _g)\) 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 \(\mathfrak B(H)\) surjects onto an appropriate symmetric Thompson group of Farley–Hughes; in particular, it answers positively. Our main result asserts that \(\mathfrak B(H)\) is of type \(F_n\) if and only if H is. As a consequence, for every \(g\in \mathbb N \cup \{0, \infty \}\) and every \(n\ge 1\) , we construct a subgroup \(G <{{\,\textrm{Map}\,}}(\Sigma _g)\) 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.
