Exponent equations in HNN-extensions

We consider exponent equations in finitely generated groups. These are equations, where the variables appear as exponents of group elements and take values from natural numbers. Solvability such (systems of) has been intensively studied for various classes groups recent years. In many cases, it turns out that set all solutions on an equation is a semilinear can be constructed effectively. Such called knapsack semilinear. Examples hyperbolic groups, virtually special co-context-free free solvable Moreover, semilinearity preserved by theoretic constructions, e.g., finite extensions, graph products, wreath amalgamated products with subgroups, HNN-extensions associated subgroups. On other hand, arbitrary do not preserve semilinearity. this paper, we HNN-extensions, stable letter $t$ acts trivially conjugation subgroup $A$ base $G$. show under some additional technical conditions, transfers $G$ to HNN-extension $H$. conditions satisfied when centralizer or quasiconvex