Authors: Ian J Leary and M\"uge Saadeto\u{g}lu 

Ian Leary
Affiliation: The Ohio State University and University of Southampton 

M\"uge Saadeto\u{g}lu
Affiliation:  University of Southampton 

Abstract: K. S. Brown defined a group $G$ to be FHT if $G$ is of 
finite virtual cohomological dimension and has the property that 
the homology groups $H_i(G;M)$ are finitely generated, for each 
$G$-module $M$ whose underlying abelian group is finitely generated.
J.-P. Serre defined $G$ to be $FP_n$ if the trivial $G$-module 
admits a projective resolution in which the terms out to degree $n$ 
are finitely generated.  
For each $n\geq 0$ we construct a torsion-free group that is FHT, 
is $FP_n$ and is not $FP_{n+1}$.  

Status: preprint