2,737 research outputs found

    Embedding universal covers of graph manifolds in products of trees

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    We prove that the universal cover of any graph manifold quasi-isometrically embeds into a product of three trees. In particular we show that the Assouad-Nagata dimension of the universal cover of any closed graph manifold is 3, proving a conjecture of Smirnov.Comment: 3 pages, final version - to appear in Proceedings of the AM

    Poorly connected groups

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    We investigate groups whose Cayley graphs have poor\-ly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini--Schramm--Tim\'ar if and only if it is virtually free. We then prove a gap theorem for connectivity of finitely presented groups, and prove that there is no comparable theorem for all finitely generated groups. Finally, we formulate a connectivity version of the conjecture that every group of type FF with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.Comment: 14 pages. Changes to v2: Proof of the Theorem 1.2 shortened, Theorem 1.4 added completing the no-gap result outlined in v

    Orthogonal forms of Kac--Moody groups are acylindrically hyperbolic

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    We give sufficient conditions for a group acting on a geodesic metric space to be acylindrically hyperbolic and mention various applications to groups acting on CAT(00) spaces. We prove that a group acting on an irreducible non-spherical non-affine building is acylindrically hyperbolic provided there is a chamber with finite stabiliser whose orbit contains an apartment. Finally, we show that the following classes of groups admit an action on a building with those properties: orthogonal forms of Kac--Moody groups over arbitrary fields, and irreducible graph products of arbitrary groups - recovering a result of Minasyan--Osin.Comment: 20 pages, to appear in Annales de l'Institut Fourie

    Poincar\'e profiles of groups and spaces

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    We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincar\'{e} profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Tim\'{a}r. In this paper we focus on properties of the Poincar\'{e} profiles of groups with polynomial growth, and of hyperbolic spaces, where we deduce a connection between these profiles and conformal dimension. As applications, we use these invariants to show the non-existence of coarse embeddings in a variety of examples.Comment: 55 pages. To appear in Revista Matem\'atica Iberoamerican

    Frammento sul male

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    Ma propre vie

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    A continuum of expanders

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    A regular equivalence between two graphs Γ,Γ\Gamma,\Gamma' is a pair of uniformly proper Lipschitz maps VΓVΓV\Gamma\to V\Gamma' and VΓVΓV\Gamma'\to V\Gamma. Using separation profiles we prove that there are 202^{\aleph_0} regular equivalence classes of expander graphs, and of finitely generated groups with a representative which isometrically contains expanders.Comment: 11 pages, accepted for publication in Fundamenta Mathematica
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