Abstract
A knapsack equation in a group G is an equation of the form g1x1⋯gkxk=g where g1,…,gk,g are elements of G and x1,…,xk are variables that take values in the natural numbers. We study the class of groups G for which all knapsack equations have effectively semilinear solution sets. We show that the following group constructions preserve effective semilinearity: graph products, amalgamated free products with finite amalgamated subgroups, HNN-extensions with finite associated subgroups, and finite extensions. Moreover, we study a complexity measure, called magnitude, of the resulting semilinear solution sets. More precisely, we are interested in the growth of the magnitude in terms of the length of the knapsack equation (measured in number of generators). We investigate how this growth changes under the above group operations.
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