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