Closure properties of knapsack semilinear groups

Abstract

We show that the following group constructions preserve the semilinearity of the solution sets for knapsack equations (equations of the form $g_1^{x_1}\cdots g_k^{x_k}=g$ in a group $G$, where the variables $x_1,\dots ,x_k$ take values in the natural numbers): graph products, amalgamated free products with finite amalgamated subgroups, HNN-extensions with finite associated subgroups, and finite extensions. Moreover, we study the dependence of the so-called magnitude for the solution set of a knapsack equation (the magnitude is a complexity measure for semi-linear sets) with respect to the length of the knapsack equation (measured in number of generators). We investigate, how this dependence changes under the above group operations.