JournalsggdOnline First1 July 2022

Equations in acylindrically hyperbolic groups and verbal closedness

  • Oleg Bogopolski

    University of Szczecin, Poland; and Heinrich-Heine-Universität Düsseldorf, Germany
Equations in acylindrically hyperbolic groups and verbal closedness cover
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Abstract

Let HH be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system SS of equations with constants from HH is equivalent to a single equation. We also show that the algebraic set associated with SS is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such an equation has the form w(x1,,xn)=hw(x_1,\ldots,x_n)=h, where wF(X)w\in F(X), hHh\in H).

From this we deduce the following statement: Let GG be an arbitrary overgroup of the above group HH. Then HH is verbally closed in GG if and only if it is algebraically closed in GG.

These statements have interesting implications; here we give only two of them: If HH is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from HH is equivalent to a single equation. We give a positive solution to Problem 5.2 from the paper [J. Group Theory 17 (2014), 29–40] of Myasnikov and Roman’kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts.

Moreover, we describe solutions of the equation xnym=anbmx^ny^m=a^nb^m in acylindrically hyperbolic groups (AH-groups), where aa, bb are non-commensurable jointly special loxodromic elements and n,mn,m are integers with sufficiently large common divisor. We also prove the existence of special test words in AH-groups and give an application to endomorphisms of AH-groups.

Cite this article

Oleg Bogopolski, Equations in acylindrically hyperbolic groups and verbal closedness. Groups Geom. Dyn. (2022), published online first

DOI 10.4171/GGD/661