Abstract: The Houghton group H_n consists of permutations p of the set {1,...,n} x N for which there is a "shift vector" (m_1,...,m_n) such that p( i, l ) = ( i, l+m_i ) for l large enough.
K. Brown proved that H_n is of type F_{n-1} but not of type F_n.
In his PhD-thesis, Franz Degenhardt, a student of Robert Bieri, considered the following variation which is based on replacing permutations by braids: Embed the set {1,...n} x N in the plane as n parallel rows of slots indexed by N. Take two parallel copies of this punctured plane and let the group HB_n consist of those braids based on the punctures that are "combed close to infinity". (This has to be illustrated by a picture which I put on a secret web page. It shows an element of HB_3.)
Degenhardt proved that HB_n is of type F_{n-1} but not of type F_n for n=1,2,3. He constructed the low dimensional skeleta of a K( HB_n, 1 ). I will describe a cube complex upon which HB_n acts and use this action to prove Degenhardt's conjecture that
For all n, HB_n is of type F_{n-1} but not of type F_n.