Non-solvable torsion-free virtually solvable groups

TL;DR

This paper investigates the structure of torsion-free, virtually solvable groups, establishing lower bounds on their Hirsch length and identifying possible simple factors, with implications for the existence of certain Bieberbach groups.

Contribution

It proves that nonsolvable, torsion-free, virtually solvable groups must have Hirsch length at least 10 and explores the possible simple factors for certain ranges.

Findings

Groups with Hirsch length ≤13 can only have A_5 as simple factor

Possible simple factors include PSL(2,7) and SL(2,8) for Hirsch length ≥14

No known examples of such groups with Hirsch length less than 15

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 $h(S)<15$.