Exponential equations in acylindrically hyperbolic groups

2021/06/21 by Agnieszka Bier, Bier, Agnieszka, Oleg Bogopolski +1
Mathematics · Physics and Astronomy · #20F65 #20F67 #20F70 #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2106.11385

openalex publication_date 2021/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be an acylindrically hyperbolic group and E an exponential equation over G. We show that if E is solvable in G, then there exists a solution whose components, corresponding to loxodromic elements, can be linearly estimated in terms of lengths of the coefficients of E. We give a more precise answer in the case where G is a relatively hyperbolic group. Under some assumption of general character, the solvability and the search problems for exponential equations over G can be reduced to the peripheral subgroups of G.

Citations

• Knapsack in graph groups, HNN-extensions and amalgamated products
• Notes about decidability of exponential equations

Related

• Structure of solutions of exponential equations in acylindrically hyperbolic groups
• Equation Problem over central extensions of hyperbolic groups
• Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks
• Equations in acylindrically hyperbolic groups and verbal closedness
• The relative exponential growth rate of subgroups of acylindrically\n hyperbolic groups