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 $x_1, \ldots, x_k$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2019
- DOI:
- 10.48550/arXiv.1911.12857
- arXiv:
- arXiv:1911.12857
- Bibcode:
- 2019arXiv191112857F
- Keywords:
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- Mathematics - Group Theory;
- 20F10;
- 20F67