Equations in acylindrically hyperbolic groups and verbal closedness
نویسندگان
چکیده
Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any system $S$ of equations with constants from is equivalent to a single equation. also the algebraic set associated is, up conjugacy, projection splitted equation (such has form $w(x\_1,\ldots,x\_n)=h$, where $w\in F(X)$, $h\in H$). From this we deduce following statement: $G$ arbitrary overgroup above $H$. Then verbally closed in if and only it algebraically $G$. These statements have interesting implications; here give two them: If non-cyclic torsion-free group, then every (possibly infinite) finitely many variables positive solution Problem 5.2 paper \[J. Group Theory 17 (2014), 29–40] Myasnikov Roman’kov: Verbally subgroups groups are retracts. Moreover, describe solutions $x^ny^m=a^nb^m$ (AH-groups), $a$, $b$ non-commensurable jointly special loxodromic elements $n,m$ integers sufficiently large common divisor. prove existence test words AH-groups application endomorphisms AH-groups.
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ژورنال
عنوان ژورنال: Groups, Geometry, and Dynamics
سال: 2022
ISSN: ['1661-7207', '1661-7215']
DOI: https://doi.org/10.4171/ggd/661