Surface Houghton groups (Q6583574)

For an integer \(n\geq 2\), first glue a torus with two boundary components (= pieces) to every boundary component of a sphere with \(n\) boundary components and then inductively glue a piece to each boundary component. This way one gets a connected orientable surface \(\Sigma_n\) with exactly \(n\) ends all of which are non-planar. The surface Houghton group, \(B_n\), is the subgroup of the mapping class group of \(\Sigma_n\) whose elements eventually send pieces to pieces, in a trivial manner.\N\NA group is called of type \(F_d\) if it has a classifying space with finite \(d\)-skeleton and that of type \(FP_d\) if the integers, regarded as a trivial module over the group, have a projective resolution that is of finite type in dimensions up to \(d\). It follows that if a group is \(F_d\) then it also \(FP_d\).\N\NThe main result of this paper proves that the group \(B_n\) is of type \(F_{n-1}\) but not of type \(FP_n\). This result is the analog of the result of \textit{A. Genevois} et al. [Geom. Topol. 26, No. 3, 1385--1434 (2022; Zbl 1507.20022)] about the braided Houghton group.