Non-solvable torsion-free virtually solvable groups

Abstract

There are perfect Bieberbach groups of Hirsch length 15, but none in lower dimensions. We shall show that a nonsolvable, torsion free, virtually solvable group $S$ must have Hirsch length $h(S)\geq10$. If $h(S)\leq13$ then we may assume that $A_5$ is the only simple factor, but $PSL(2,7)$ and $SL(2,8)$ may occur when $h(S)\geq14$. There are no known examples with $h(S)<15$.