On finite systems of equations in acylindrically hyperbolic groups

2019/03/24 by Bogopolski, Oleg
#20F65 #20F67 #20F70 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1903.10906

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(x₁,…,x_n)=h, where w∈ F(X), h∈ 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.

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