Finiteness properties of arithmetic groups over function fields

In joint work with Kai-Uwe Bux, we show that an arithmetic
subgroup of a reductive group defined over a global function field is of
type $FP_\infty$ if and only if the semisimple rank of the reductive group
over the global field equals 0.

For example, $SL(n,F[t])$ is not of type $FP_\infty$ for any finite field F.

This result confirms a conjecture from the 1970’s that had its roots in
the work of Serre and Stuhler. Our proof is motivated by the
Epstein-Thurston proof that $SL(n,Z)$ is not combable when $n>2$.