532 research outputs found

    Combination of convergence groups

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    We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively hyperbolic groups over a fully quasi-convex subgroup. We apply our result to Sela's theory on limit groups and prove their relative hyperbolicity. We also get a proof of the Howson property for limit groups.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper27.abs.htm

    Makanin-Razborov diagrams for limit groups

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    We give a description of Hom(G,L)Hom(G,L), where LL is a limit group (fully residually free group). We construct a finite diagram of groups, Makanin-Razborov diagram, that gives a convinient representation of all such homomorphisms.Comment: 22 pages, 7 figure

    A Jorgensen-Thurston theorem for homomorphisms

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    In this note, we provide a description of the structure of homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jorgensen-Thurston Theorem in hyperbolic geometry.Comment: 16 pages, 4 figure

    A characterization of higher rank symmetric spaces via bounded cohomology

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    Let MM be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group Γ\Gamma does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover M~\tilde M is a higher rank symmetric space iff Hb2(M;R)H2(M;R)H^2_b(M;\R)\to H^2(M;\R) is injective (and otherwise the kernel is infinite-dimensional). This is the converse of a theorem of Burger-Monod. The proof uses the celebrated Rank Rigidity Theorem, as well as a new construction of quasi-homomorphisms on groups that act on CAT(0) spaces and contain rank 1 elements
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