Exponent equations in HNN-extensions

Abstract

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 HNNextensions 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.This work has been supported by the DFG research project LO 748/13-1. 1 Many groups are known to be virtually special, e.g., Coxeter groups, fully residually free groups, one-relator groups with torsion, and fundamental groups of hyperbolic 3-manifolds.

Exponent Equations in HNN-extensions

“…This leads to several cases. A systematic consideration of all cases can be found in [9]; it follows the proof of the knapsack semilinearity of hyperbolic groups from [23]. Here, we only want to consider one typical case: Assume there is a path 𝑃 of length at most 𝜅 connecting a point 𝑎 on 𝑃 2 (1 ≤ 𝑖 ≤ 𝑛) with a point on 𝑄 𝑖 with 3 ≤ 𝑖 ≤ 𝑛.…”