Growth of associated monomial algebras with application to Manturov groups

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.