Short positive loxodromics in graph products

TL;DR

This paper introduces a method to generate loxodromic elements in graph product groups using positive words, impacting subgroup growth and properties of growth rates.

Contribution

It provides a new technique for constructing loxodromics and explores implications for subgroup growth and the ordering of growth rates in graph products.

Findings

Effective method for generating loxodromics in graph products

Graph products with strong growth properties inherit those properties

Set of growth rates is well-ordered in certain subgroup classes

Abstract

We give a method for effectively generating generalised loxodromics in subgroups of graph products, using positive words. This has several consequences for the growth of subsets of these groups. In particular, we show that graph products of groups with strong product set growth properties also share those properties. We additionally show that the set of growth rates of a class of subgroups of any graph product of equationally noetherian groups is well-ordered.