Structure of solutions of exponential equations in acylindrically hyperbolic groups

2022/05/23 by Bier, Agnieszka, Bogopolski, Oleg
#20F65 #20F67 (Secondary) #20F70 (Primary) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2205.11662

Abstract

Let G be a group acting acylindrically on a hyperbolic space and let E be an exponential equation over G. We show that E is equivalent to a finite disjunction of finite systems of pairwise independent equations which are either loxodromic over virtually cyclic subgroups or elliptic. We also obtain a description of the solution set of E. We obtain stronger results in the case where G is hyperbolic relative to a collection of peripheral subgroups \H_λ\_{λ∈ Λ}. In particular, we prove in this case that the solution sets of exponential equations over G are ℤ-semilinear if and only if the solution sets of exponential equations over every H_λ, λ∈ Λ, are ℤ-semilinear. We obtain an analogous result for finite disjunctions of finite systems of exponential equations and inequations over relatively hyperbolic groups in terms of definable sets in the weak Presburger arithmetic.

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