Abstract
We prove that every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient. As an application, we obtain an acylindrically hyperbolic group Q with strong fixed point properties: Q has property FLp for all p∈[1,+∞), and every action of Q on a finite dimensional contractible topological space has a fixed point. In addition, Q has other properties which are rather unusual for groups exhibiting “hyperbolic-like” behavior. E.g., Q is not uniformly non-amenable and has finite generating sets with arbitrary large balls consisting of torsion elements.
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