On the knapsack problem and semilinear sets

In this thesis we analyze the knapsack problem for different group constructions and groups. The knapsack problem, which has become well known in optimization and economics, is considered here in a group-theoretic context as a decision problem.
Of particular interest to us are groups in which the set of solution vectors forms a so-called semilinear set for each input. Such groups are called knapsack-semilinear. The set of knapsack-semilinear groups satisfies good closure properties, some of which we discuss in this thesis.
We define the concept of a magnitude and determine in the first part the magnitude of knapsack-semilinear groups under finite extensions, graph products and HNN-extensions or amalgamated products with certain restrictions. It turns out that solvability of knapsack equations of such group constructions is in NP if this is already the case for the base groups.
We then show that certain HNN-extensions of knapsack-semilinear groups over infinite associated subgroups are also knapsack-semilinear, if we restrict ourselves to the case where the isomorphism between the subgroups is the identity. An important special case here are the so-called extensions of centralizers. The same applies to central extensions of hyperbolic groups: These are also knapsack-semilinear. As an application, we then conclude that HNN-extensions (of the mentioned restricted form) of hyperbolic groups over quasiconvex subgroups are knapsack-semilinear.
In the last part of the thesis we consider the knapsack problem for two more cases, but not from the semilinear aspect. For uniformly SENS groups G the knapsack problem for G ≀ Z is hard in the second existential level of the polynomial time hierarchy. Furthermore, we show that the knapsack problem for SL₃(Z) is already undecidable in the case of a single exponent equation (where variables can occur multiple times).