# On finite systems of equations in acylindrically hyperbolic groups

## TL;DR

This paper proves that in acylindrically hyperbolic groups without finite normal subgroups, finite systems of equations can be simplified to a single equation, revealing structural properties of algebraic sets and subgroup closures.

## Contribution

It establishes the equivalence of finite systems of equations to a single equation in acylindrically hyperbolic groups and explores implications for algebraic and verbal closure properties.

## Key findings

- Finite systems of equations are equivalent to a single equation in the specified groups.
- Algebraic sets associated with systems are projections of sets from single splitted equations.
- H is verbally closed in G if and only if it is algebraically closed in G.

## Abstract

Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system $S$ of equations with constants from $H$ is equivalent to a single equation. We also show that the algebraic set associated with $S$ is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such equation has the form $w(x_1,\dots,x_n)=h$, where $w\in F(X)$, $h\in H$). From this we deduce the following statement: Let $G$ be an arbitrary overgroup of the above group $H$. Then $H$ is verbally closed in $G$ if and only if it is algebraically closed in $G$. Another corollary: If $H$ is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from $H$ is equivalent to a single equation.