Virtually torsion-free covers of minimax groups

TL;DR

This paper shows that finitely generated virtually solvable minimax groups can be covered by virtually torsion-free groups, enabling generalizations of random walk theorems and identifying preserved properties.

Contribution

It introduces a method to obtain virtually torsion-free covers for finitely generated virtually solvable minimax groups and extends related theorems.

Findings

Finitely generated virtually solvable minimax groups have virtually torsion-free covers.

Properties like derived length and nilpotency class are preserved in the cover.

Characterization of infinitely generated virtually solvable minimax groups with such covers.

Abstract

We prove that every finitely generated, virtually solvable minimax group can be expressed as a homomorphic image of a virtually torsion-free, virtually solvable minimax group. This result enables us to generalize a theorem of Ch. Pittet and L. Saloff-Coste about random walks on finitely generated, virtually solvable minimax groups. Moreover, the paper identifies properties, such as the derived length and the nilpotency class of the Fitting subgroup, that are preserved in the covering process. Finally, we determine exactly which infinitely generated, virtually solvable minimax groups also possess this type of cover.