Michael Figelius ; Markus Lohrey - Exponent equations in HNN-extensions

gcc:10078 - journal of Groups, complexity, cryptology, December 26, 2022, Volume 14, Issue 2 - https://doi.org/10.46298/jgcc.2022.14.2.10521
Exponent equations in HNN-extensionsArticle

Authors: Michael Figelius ORCID1,2; Markus Lohrey ORCID1,2

  • 1 Department of Electrical Engineering and Computer Science, University of Siegen, Germany
  • 2 University of Siegen

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. Examples of knapsack semilinear groups are hyperbolic groups, virtually special groups, co-context-free groups and free solvable groups. Moreover, knapsack semilinearity is preserved by 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 HNN-extensions 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 base group $G$ to the HNN-extension $H$. 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 $G$.

Volume: Volume 14, Issue 2
Published on: December 26, 2022
Accepted on: December 22, 2022
Submitted on: September 26, 2022
Keywords: Mathematics - Group Theory,20F10, 20F67

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