Finitely generated groups with polynomial index growth

TL;DR

This paper characterizes finitely generated soluble residually finite groups with polynomial index growth as minimax groups and shows such groups are linear if residually finite-soluble, with implications for boundedly generated groups.

Contribution

It establishes a precise equivalence between polynomial index growth and being a minimax group for finitely generated soluble residually finite groups, and links PIG groups to linearity.

Findings

Finitely generated soluble residually finite groups with PIG are minimax.

Residually finite-soluble PIG groups are linear.

Infinite boundedly generated residually finite groups have infinite linear quotients.

Abstract

We prove that a finitely generated soluble residually finite group has polynomial index growth if and only if it is a minimax group. We also show that if a finitely generated group with PIG is residually finite-soluble then it is a linear group. These results apply in particular to boundedly generated groups; they imply that every infinite BG residually finite group has an infinite linear quotient.