Return to Revisions

Here's a result that gives some idea of how hard it is to characterise linear (let alone residually finite) groups of type $F$ (ie with a $K(G,1)$ that's a finite complex).

Theorem: There is a sequence of finite subsets $S_i\subseteq GL_{n_i}(\mathbb{Z})$ with the property that:

1. for every $i$, either $G_i=\langle S_i\rangle$ is of type $F$ or $G_i$ is not finitely presentable (in particular not of type $F$);
2. the set of $i$ such that $G_i$ is of type $F$ is recursively enumerable but not recursive.

So there is no algorithm to determine whether or not $G_i$ is of type $F$.

I can give details of the proof if anyone's interested. Basically, it's an easy application of the Haglund--Wise version of the Rips Construction.