Images of Golod-Shafarevich algebras with small growth

TL;DR

This paper demonstrates that Golod-Shafarevich algebras can be mapped onto infinite-dimensional algebras with controlled polynomial or quadratic growth, answering a question by Zelmanov and constructing algebras with prescribed growth rates.

Contribution

It shows how Golod-Shafarevich algebras can be homomorphically mapped onto algebras with small growth, including quadratic, under mild conditions, and constructs algebras with specific growth functions.

Findings

Golod-Shafarevich algebras can map onto infinite-dimensional algebras with polynomial growth

Finitely presented Golod-Shafarevich algebras can map onto quadratic growth algebras

Any sufficiently regular function at least n^{log n} can be realized as an algebra's growth

Abstract

We show that Golod-Shafarevich algebras can be homomorphically mapped onto infinite-dimensional algebras with polynomial growth, under mild assumptions of the number of relations of given degrees. In case these algebras are finitely presented, we show they can be mapped onto an infinite dimensional algebras with quadratic growth. This answers a guestion by Zelmanov. We then show, by an elementary construction, that any sufficiently regular function at least $n^{\log n}$ may occur as the growth of an algebra.