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 $h(S)\ge 10$. If $h(S)\le 13$ then we may assume that $A_5$ is the only simple factor, but $PSL(2,7)$ and $SL(2,8)$ may occur when $h(S)\ge 14$. There are no known examples with $h(S)<15$.

v2: Results in \S5 and \S7 sharpened. Reference [5] updated. v3: New \S8, using torsion in crystallographic quotients. v4:Theorem 20 corrected, \S8 expanded. v5: Lemmas 4 and 7, and Sections 5 and 8 rewritten. v6: Reorganised. v: PSL(2,7) and SL(2,8) excluded in dimensions $<14$