Equations in acylindrically hyperbolic groups and verbal closedness

Groups, geometry, and dynamics

article

Oleg Bogopolski¹ 

[1] University of Szczecin;Heinrich-Heine-Universität Düsseldorf

关键词: Equations over a group;    acylindrically hyperbolic group;    algebraically closed subgroup;    verbally closed subgroup;    retract;    relatively hyperbolic group;    equationally Noetherian group;   

DOI  :  10.4171/ggd/661

学科分类：神经科学

来源: European Mathematical Society

【 摘 要 】

Let HHH be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system SSS of equations with constants from HHH is equivalent to a single equation. We also show that the algebraic set associated with SSS 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)=hw(x1​,…,xn​)=h, where w∈F(X)w\in F(X)w∈F(X), h∈Hh\in Hh∈H). From this we deduce the following statement: Let GGG be an arbitrary overgroup of the above group HHH. Then HHH is verbally closed in GGG if and only if it is algebraically closed in GGG. These statements have interesting implications; here we give only two of them: If HHH is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from HHH 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^mxnym=anbm in acylindrically hyperbolic groups (AH-groups), where aaa, bbb are non-commensurable jointly special loxodromic elements and n,mn,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.

【 授权许可】

CC BY