Exponent equations in HNN-extensions

TL;DR

This paper investigates when the property of knapsack semilinearity in groups is preserved in HNN-extensions, especially those where the stable letter acts trivially on the associated subgroup, extending known results in group theory.

Contribution

It establishes conditions under which knapsack semilinearity transfers from a base group to certain HNN-extensions, broadening understanding of this property in complex group constructions.

Findings

Knapsack semilinearity is preserved under specific conditions in HNN-extensions.

The property holds when the associated subgroup is a centralizer or quasiconvex in a hyperbolic group.

The paper extends previous results on semilinearity to a wider class of HNN-extensions.

Abstract

We consider exponent equations in finitely generated groups. These are equations, where the variables appear as exponents of group elements and take values from the natural numbers. Solvability of such (systems of) equations has been intensively studied for various classes of groups in recent years. In many cases, it turns out that the set of all solutions on an exponent equation is a semilinear set that can be constructed effectively. Such groups are called knapsack semilinear. Examples of knapsack semilinear groups are hyperbolic groups, virtually special groups, co-context-free groups and free solvable groups. Moreover, knapsack semilinearity is preserved by many group theoretic constructions, e.g., finite extensions, graph products, wreath products, amalgamated free products with finite amalgamated subgroups, and HNN-extensions with finite associated subgroups. On the other hand, arbitrary HNN-extensions do not preserve knapsack semilinearity. In this paper, we consider the knapsack semilinearity of HNN-extensions, where the stable letter $t$ acts trivially by conjugation on the associated subgroup $A$ of the base group $G$. We show that under some additional technical conditions, knapsack semilinearity transfers from base group $G$ to the HNN-extension $H$. These additional technical conditions are satisfied in many cases, e.g., when $A$ is a centralizer in $G$ or $A$ is a quasiconvex subgroup of the hyperbolic group $G$.