Exponent equations in HNN-extensions (Q6601462)

Let \(G\) be a finitely generated (infinite) group. An exponent equation over \(G\) has the form \(h_{0}g_{0}^{x_{0}}h_{1}g_{1}^{x_{1}} \ldots h_{n}g_{n}^{x_{n}}h_{n+1}=1\), where \(g_{i}, h_{i} \in G\) and \(x_{i}\) are variables that range over \(\mathbb{N}\), the set of non-negative integers. Solvability of such (systems of) equations has been intensively studied for various classes of groups. If the set of all solutions on an exponent equation over \(G\) is a semilinear set that can be constructed effectively, then \(G\) is called knapsack semilinear. In [\textit{M. Figelius} ez al., J. Algebra 589, 437--482 (2022; Zbl 1512.20105)] it was shown that the class of knapsack semilinear groups is closed under finite extensions, graph 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.\N\NIn the paper under review, the authors consider the knapsack semilinearity of HNN-extensions of the form \(H= \langle G,t \mid a^{t}=a, a \in A \rangle\), where the stable letter \(t\) acts trivially by conjugation on the associated subgroup \(A \leq G\). They 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, in particular when \(A=C_{G}(S)\) is the centralizer in \(G\) of a finite subset \(S \subset G\) (see Theorem 5.7), or \(A\) is a quasiconvex subgroup of the hyperbolic group \(G\) (see Theorem 6.8).