On the Gap Conjecture concerning group growth

TL;DR

This paper explores the Gap Conjecture on group growth, reducing its proof to specific classes of groups such as simple, residually finite, and residually polycyclic groups, thereby narrowing the scope of verification needed.

Contribution

It provides new reductions of the Gap Conjecture to particular classes of groups, simplifying the problem by focusing on these subclasses.

Findings

Validity for simple and residually finite groups implies the conjecture generally.

Reductions also hold for residually polycyclic and just-infinite groups.

The paper discusses cases for residually solvable and right orderable groups.

Abstract

We discuss some new results concerning Gap Conjecture on group growth and present a reduction of it (and its *-version) to several special classes of groups. Namely we show that its validity for the classes of simple groups and residually finite groups will imply the Gap Conjecture in full generality. A similar type reduction holds if the Conjecture is valid for residually polycyclic groups and just-infinite groups. The cases of residually solvable groups and right orderable groups are considered as well.