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Abstract We prove that many relatively hyperbolic groups obtained by relative strict hyperbolization admit a cocompact action on a cubical complex. Under suitable assumptions on the peripheral subgroups, these groups are residually finite and even virtually special. We include some applications to the theory of manifolds, such as the construction of new non‐positively curved Riemannian manifolds with residually finite fundamental group, and the existence of non‐triangulable aspherical manifolds with virtually special fundamental group.
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Abstract We prove that the Gromov hyperbolic groups obtained by the strict hyperbolization procedure of Charney and Davis are virtually compact special, hence linear and residually finite. Our strategy consists in constructing an action of a hyperbolized group on a certain dual$$\operatorname {CAT}(0)$$ cubical complex. As a result, all the common applications of strict hyperbolization are shown to provide manifolds with virtually compact special fundamental group. In particular, we obtain examples of closed negatively curved Riemannian manifolds whose fundamental groups are linear and virtually algebraically fiber.more » « less
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We consider finite $$2$$ -complexes $$X$$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT( $-1$ ) metrics on $$X$$ , which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices, are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $$X$$ . As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $$X$$ .more » « less
