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Geometric intersection number and analogues of the curve complex for free groups

Geometric intersection number and analogues of the curve complex for free groups,10.2140/gt.2009.13.1805,Geometry & Topology,Ilya Kapovich,Martin Lust

Geometric intersection number and analogues of the curve complex for free groups   (Citations: 9)
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Journal: Geometry & Topology - GEOM TOPOL , vol. 13, no. 3, pp. 1805-1833, 2009
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    • ...The intersection form was introduced in [Ka3,4], [Lu] for free simplicial actions of F , that is for the non-projectivized outer space cv(F ). In a recent paper [KaLu2] we proved that the intersection form extends continuously to the closure cv(F )o fcv(F ) consisting of all minimal very small isometric actions of F on R-trees...
    • ...Ursula Hamenst¨ adt [H] recently used our result from [KaLu2] about the continuous extension of the intersection form to cv(F ) as a key ingredient to prove that any non-elementary subgroup of Out(F ), where N ≥ 3, has infinite-dimensional second bounded cohomology group (infinite-dimensional space of quasi-morphisms)...
    • ...Very recently Bestvina and Feighn [BeF2] used [KaLu2] to show that for any finite collection φ1 ,...,φ m ∈ Out(FN )o fiwip outer automorphisms of FN (“irreducible automorphisms with irreducible powers”, see Definition 12.1) there exists a δ-hyperbolic complex X = X(φ1 ,...,φ m )w ith an isometric Out( FN )-action where each φi acts with a positive asymptotic translation length...
    • ...If Tλ denotes the “dual” R-tree transverse to λ with metric defined by the transverse measure on λ (see Ch. 11.12 in [Kap] for details), then the definition in [KaLu2] gives...
    • ...One of the main motivations and prospective uses for Theorem 1.1 is to analyze the intersection graph, introduced by the authors in [KaLu2] in order to study various free group analogues of the curve complex...
    • ...Thus one can define a graph, whose vertices are conjugacy classes of primitive elements in F where two vertices [a], [b ]a re adjacent if there existsT ∈ cv(F ) such that � T, ηa� = � T, ηb� =0 , that is � a� T = � b� T = 0 (this graph is almost the same as the “dual cut graph” defined in [KaLu2])...
    • ...This leads to the notion [KaLu2] of a cut graph for F whose vertices are nontrivial splittings of F as the fundamental group of a graph of groups with a single edge and the trivial edge group, and where adjacency again corresponds to having a common refinement...
    • ...In [KaLu2] we prove that for N ≥ 3 the intersection graph and all the free group analogues of the curve complex derived from it have infinite diameter, by analyzing the action of iwip automorphisms (see Definition 12.1)...
    • ...In the terminology of [KaLu2], the assumption of Theorem 1.4 says that the distance between [T1 ]a nd [ T2] in the intersection graph I(F ) is bigger than two...
    • ...Recently, Kapovich and Lustig [KaLu2] generalized this result to the case of arbitrary very small actions and proved Proposition–Definition 2.11 in the form stated above...
    • ...Therefore, by the continuity of the intersection form [KaLu2], we have...
    • ...By the continuity of the intersection form on the closure of the non-projectivized outer space (see [KaLu2]), this implies that...

    Ilya Kapovichet al. Intersection Form, Laminations and Currents on Free Groups

    • ...A discussion of some of these analogs and their basic properties is provided in [23]...
    • ...By analogy with the curve complex situation where pseudo-Anosov homeomorphisms have unbounded orbits and other homeomorphisms have bounded orbits, Kapovich and Lustig have shown that fully irreducible automorphisms act with unbounded orbits and other automorphisms with bounded orbits [23]...

    Jason Behrstocket al. Growth of intersection numbers for free group automorphisms

    • ...It should be remarked that different notions of intersection number have been developed by Scott–Swarup [SS], Guirardel [Gu2], and Kapovich–Lustig [KL3], but that ours has been tailored to suit the needs of our theorem...
    • ...To build filling cyclic trees for arbitrarily high rank free groups we introduce two simplicial complexes naturally associated to Fk; these complexes appear in [KL3]...
    • ...is the following variant of the like-named complex appearing in [KL3]: Vertices correspond to very small simplicial trees for Fk, i.e...
    • ...Remark 2.9. For k ≥ 3, Kapovich and Lustig have shown that for a hyperbolic fully irreducible element φ ∈ Out Fk and any two vertices [A], [B] ∈D that dD([A] ,φ n ([B])) goes to infinity as n →± ∞([KL3])...

    Matt Clayet al. Twisting Out Fully Irreducible Automorphisms

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