Growth of associated monomial algebras with application to Manturov groups

TL;DR

This paper explores the relationship between the Gelfand-Kirillov dimension of associative algebras and their monomial counterparts, providing conditions for equality and applying results to analyze the growth of Manturov groups, revealing diverse growth behaviors.

Contribution

It establishes sufficient conditions under which an algebra and its monomial algebra share the same GK-dimension, extending previous results and applying them to Manturov groups.

Findings

Manturov (1,n)-groups have zero growth for n>1

Manturov (2,3)-group has quadratic growth

Manturov (k,n)-groups with n>k≥3 contain free subgroups and have exponential growth

Abstract

It is well-known that an associative algebra shares the same growth and Gelfand-Kirillov dimension (GK-dimension) as its associated monomial algebra with respect to a degree-lexicographic order. This article mainly investigates the relationship between the GK-dimension of an algebra and that of its associated monomial algebra with respect to a monomial order. We obtain sufficient conditions on a monomial order such that these two algebras have the same GK-dimension. Our result generalizes the well-known result and has several applications. In particular, as an application, we study the growth of Manturov $(k,n)$-groups for positive integers $n>k$. It is shown that the Manturov $(1,n)$-group has growth equal to $0$ for all $n>1$; the Manturov $(2,3)$-group has growth equal to $2$; and, for all $n>k\geq3$, the Manturov $(k,n)$-group contains a free subgroup of rank $2$ and thus has exponential growth.