Non-solvable torsion-free virtually solvable groups

Jonathan A. Hillman

Published 2023-02-19Version 1

We show that a non-solvable, torsion-free, virtually solvable group $S$ must have Hirsch length $h(S)\geq7$ and either be virtually nilpotent and of nilpotency class $\leq3$ or have $h(S)\geq8$. If $S$ is virtually polycyclic but not virtually nilpotent then $h(S)\geq9$. (There are virtually abelian examples with Hirsch length 15, and this is known to be best possible in the virtually abelian case.)