ℝ-trees and laminations for free groups II: the dual lamination of an ℝ-tree

1805Geometric intersection number and analogues of the curve complex for free groups ILYA KAPOVICH MARTIN LUSTIGFor the free group F N of finite rank N 2 we construct a canonical Bonahon-type, continuous and Out.F N /-invariant geometric intersection formHere cv.F N / is the closure of unprojectivized Culler-Vogtmann Outer space cv.F N / in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that cv.F N / consists of all very small minimal isometric actions of F N on R-trees. The projectivization of cv.F N / provides a free group analogue of Thurston's compactification of Teichmüller space.As an application, using the intersection graph determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter.

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“…The following is an easy corollary of the definitions (see Lemma 3.1(b) of [15] or Lemma 3.1 of [11]):…”

Section: Definition 31 (Bounded Back-tracking Constant)mentioning

confidence: 99%

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

Kapovich

Lustig²

2009

Geom. Topol.

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144

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“…It is not hard to show (see [LevL], [CouHL2]) that this definition of L 2 (T ) does not depend on the choice of a simplicial chart on F .…”

Section: Definition 31 (Algebraic Laminations) An Algebraic Laminatmentioning

confidence: 99%

“…Proposition-Definition 10.1 [CouHL2]. Let F be a nonabelian finitely generated group with a very small minimal isometric action on an R-tree T .…”

Section: The General Casementioning

confidence: 99%

“…We need the following basic fact about the structure of a general very small action (see Remark 2.1 in [CouHL2]). …”

Section: The General Casementioning

confidence: 99%

“…In particular, given a very small action of F on an R-tree T , there is [CouHL2] a naturally defined "dual lamination" L 2 (T ) on F (see section 3 below). In the special case where T belongs to cv (F ), i.e.…”

Section: Introductionmentioning

confidence: 99%

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Intersection Form, Laminations and Currents on Free Groups

Kapovich

Lustig

2009

Geom. Funct. Anal.

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Abstract. Let F be a free group of rank N ≥ 2, let µ be a geodesic current on F and let T be an R-tree with a very small isometric action of F . We prove that the geometric intersection number T, µ is equal to zero if and only if the support of µ is contained in the dual algebraic lamination L 2 (T ) of T . Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. We use the main result to obtain "unique ergodicity" type properties for the attracting and repelling fixed points of atoroidal iwip elements of Out(F ) when acting both on the compactified outer Space and on the projectivized space of currents. We also show that the sum of the translation length functions of any two "sufficiently transverse" very small F -trees is bilipschitz equivalent to the translation length function of an interior point of the outer space. As another application, we define the notion of a filling element in F and prove that filling elements are "nearly generic" in F . We also apply our results to the notion of bounded translation equivalence in free groups.

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On indecomposable trees in the boundary of outer space

Reynolds

2010

Geom Dedicata

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Abstract. Let T be an R-tree, equipped with a very small action of the rank n free group Fn, and let H ≤ Fn be finitely generated. We consider the case where the action Fn T is indecomposable-this is a strong mixing property introduced by Guirardel. In this case, we show that the action of H on its minimal invarinat subtree T H has dense orbits if and only if H is finite index in Fn. There is an interesting application to dual algebraic laminations; we show that for T free and indecomposable and for H ≤ Fn finitely generated, H carries a leaf of the dual lamination of T if and only if H is finite index in Fn. This generalizes a result of Bestvina-Feighn-Handel regarding stable trees of fully irreducible automorphisms.

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ℝ-trees and laminations for free groups II: the dual lamination of an ℝ-tree

Cited by 55 publications

References 23 publications

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

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

Intersection Form, Laminations and Currents on Free Groups

On indecomposable trees in the boundary of outer space

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