Exponent Equations in HNN-extensions

Michael Figelius, Markus Lohrey
2022 Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation  
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. The class of
more » ... psack semilinear groups is quite rich and it is closed under 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 HNNextensions 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 the base group 𝐺 to the HNN-extension. 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. CCS CONCEPTS • Theory of computation → Formal languages and automata theory.
doi:10.1145/3476446.3535482 fatcat:3tapsx3hdbcyfa27ws3kxhgmdy