[Paper Review] Exponential equations in acylindrically hyperbolic groups

Agnieszka Bier, Oleg Bogopolski|arXiv (Cornell University)|Jun 21, 2021

Geometric and Algebraic Topology22 references4 citations

TL;DR

This paper establishes linear bounds on solutions to exponential equations in acylindrically hyperbolic groups, particularly for loxodromic elements, by leveraging geometric group theory techniques. It proves that solvability over such groups reduces to solvability over peripheral subgroups in the relatively hyperbolic case, enabling algorithmic decision procedures under known hyperbolicity constants and recursive presentations.

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 $G$.

Motivation & Objective

• To extend decidability results for exponential equations from hyperbolic groups to the broader class of acylindrically hyperbolic groups.
• To establish effective linear bounds on solution sizes for loxodromic generators in exponential equations over acylindrically hyperbolic groups.
• To reduce the solvability and search problems for exponential equations in relatively hyperbolic groups to their peripheral subgroups under computable conditions.
• To provide an algorithmic framework for solving exponential equations in finitely generated relatively hyperbolic groups with known hyperbolicity constants and recursive peripheral presentations.

Proposed method

• Use of geometric techniques in acylindrically hyperbolic groups, particularly the action on hyperbolic spaces and the classification of group elements as elliptic, parabolic, or loxodromic.
• Application of the concept of hyperbolically embedded subgroups to reduce exponential equations to equations over peripheral subgroups.
• Construction of a finite disjunction of systems of exponential equations over peripheral subgroups, ensuring solvability equivalence.
• Computation of a uniform constant $ M $ based on group-theoretic invariants such as hyperbolicity constant $ \delta $, generating set size, and relator lengths.
• Use of bounded solution search via finite search spaces for elements of finite order and loxodromic elements with linearly bounded exponents.
• Algorithmic reduction of the original equation to a disjunction of equations over peripheral subgroups, with solvability and solution extension preserved.

Experimental results

Research questions

• RQ1 Can exponential equations in acylindrically hyperbolic groups be solved with solution sizes bounded linearly in terms of coefficient lengths?
• RQ2 Under what conditions can the solvability of an exponential equation in a relatively hyperbolic group be reduced to its peripheral subgroups?
• RQ3 Is there an algorithm to compute a finite disjunction of systems over peripheral subgroups that captures the solvability of a given exponential equation in a relatively hyperbolic group?
• RQ4 Can the solution size for loxodromic generators in exponential equations be effectively bounded in acylindrically hyperbolic groups?

Key findings

• For any acylindrically hyperbolic group $ G $ with generating set $ X $, there exists a constant $ M > 1 $ such that any solvable exponential equation has a solution with $ |k_j| \leq \left(n^2 + \sum |a_i|_X + \sum |g_i|_X\right) \cdot M $ for all loxodromic $ g_j $.
• In the relatively hyperbolic case, the solvability of an exponential equation is equivalent to the solvability of a finite disjunction of systems of equations over peripheral subgroups.
• The algorithmic reduction to peripheral subgroups is effective when the hyperbolicity constant $ \delta $, the finite relative presentation, and recursive presentations of peripheral subgroups are known.
• The constant $ M $ in the solution bound can be algorithmically computed from $ |X| $, $ \delta $, and the maximum length of relators in the relative presentation.
• For hyperbolic groups, the result implies that the solution size bound is linear, improving upon the earlier polynomial bound from Myasnikov et al.
• The example of $ H * F_2 $ with $ H $ containing $ \mathbb{Q} $ shows that loxodromic elements are essential in the bound — without them, no uniform bound exists.