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27
Geometric Intersection Number and analogues of the Curve Complex for free groups
, 2007
"... For the free group FN of finite rank N ≥ 2 we construct a canonical Bonahontype, continuous and Out(FN)invariant geometric intersection form 〈 , 〉 : cv(FN) × Curr(FN) → R≥0. Here cv(FN) is the closure of unprojectivized CullerVogtmann’s Outer space cv(FN) in the equivariant GromovHausdorff c ..."
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Cited by 42 (15 self)
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For the free group FN of finite rank N ≥ 2 we construct a canonical Bonahontype, continuous and Out(FN)invariant geometric intersection form 〈 , 〉 : cv(FN) × Curr(FN) → R≥0. Here cv(FN) is the closure of unprojectivized CullerVogtmann’s Outer space cv(FN) in the equivariant GromovHausdorff convergence topology (or, equivalently, in the length function topology). It is known that cv(FN) consists of all very small minimal isometric actions of FN on Rtrees. The projectivization of cv(FN) provides a free group analogue of Thurston’s compactification of the 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.
Stabilizers of Rtrees with free isometric actions of FN
 J. Group Theory
"... Abstract. We prove that if T is an Rtree with a minimal free isometric action of FN, then the Out(FN)stabilizer of the projective class [T] is virtually cyclic. For the special case where T = T+(ϕ) is the forward limit tree of an atoroidal iwip element ϕ ∈ Out(FN) this is a consequence of the resu ..."
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Abstract. We prove that if T is an Rtree with a minimal free isometric action of FN, then the Out(FN)stabilizer of the projective class [T] is virtually cyclic. For the special case where T = T+(ϕ) is the forward limit tree of an atoroidal iwip element ϕ ∈ Out(FN) this is a consequence of the results of Bestvina, Feighn and Handel [6], via very different methods. We also derive a new proof of the Tits alternative for subgroups of Out(FN) containing an iwip (not necessarily atoroidal): we prove that every such subgroup G ≤ Out(FN) is either virtually cyclic or contains a free subgroup of rank two. The general case of the Tits alternative for subgroups of Out(FN) is due to Bestvina, Feighn and Handel. 1.
TWISTING OUT FULLY IRREDUCIBLE AUTOMORPHISMS
, 2009
"... By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill, every element not conjugate to a power of one of the twists is pseudoAnosov. We prove an analogue of this theorem for the outer automorphism group of a free group. ..."
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By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill, every element not conjugate to a power of one of the twists is pseudoAnosov. We prove an analogue of this theorem for the outer automorphism group of a free group.
Spectral rigidity of automorphic orbits in free groups
"... It is wellknown that a point T ∈ cvN in the (unprojectivized) CullerVogtmann Outer space cvN is uniquely determined by its translation length function .T: FN → R. A subset S of a free group FN is called spectrally rigid if, whenever T, T ′ ∈ cvN are such that gT = g  T ′ for every g ..."
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Cited by 10 (3 self)
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It is wellknown that a point T ∈ cvN in the (unprojectivized) CullerVogtmann Outer space cvN is uniquely determined by its translation length function .T: FN → R. A subset S of a free group FN is called spectrally rigid if, whenever T, T ′ ∈ cvN are such that gT = g  T ′ for every g ∈ S then T = T ′ in cvN. By contrast to the similar questions for the Teichmüller space, it is known that for N ≥ 2 there does not exist a finite spectrally rigid subset of FN. In this paper we prove that for N ≥ 3 if H ≤ Aut(FN) is a subgroup that projects to an infinite normal subgroup in Out(FN) then the Horbit of an arbitrary nontrivial element g ∈ FN is spectrally rigid. We also establish a similar statement for F2 = F (a, b), provided that g ∈ F2 is not conjugate to a power of [a, b].
Rtrees, dual laminations, and compact systems of partial isometries
, 2009
"... Let FN be a free group of finite rank N ≥ 2, and let T be an Rtree with a very small, minimal action of FN with dense orbits. For any basis A of FN there exists a heart KA ⊂ T ( = the metric completion of T) which is a compact subtree that has the property that the dynamical system of partial isome ..."
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Cited by 9 (3 self)
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Let FN be a free group of finite rank N ≥ 2, and let T be an Rtree with a very small, minimal action of FN with dense orbits. For any basis A of FN there exists a heart KA ⊂ T ( = the metric completion of T) which is a compact subtree that has the property that the dynamical system of partial isometries ai: KA ∩aiKA → a −1 i KA ∩KA, for each ai ∈ A, defines a tree T(KA,A) which contains an isometric copy of T as minimal subtree. 1
Invariant laminations for irreducible automorphisms of free groups
"... For every irreducible hyperbolic automorphism ϕ of FN (i.e. the analog of a pseudoAnosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree T+(ϕ) is obtained as ‘diagonal closure ’ of the support of the backward limit current μ−(ϕ). This diagonal closure is obta ..."
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For every irreducible hyperbolic automorphism ϕ of FN (i.e. the analog of a pseudoAnosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree T+(ϕ) is obtained as ‘diagonal closure ’ of the support of the backward limit current μ−(ϕ). This diagonal closure is obtained through a finite procedure analogous to adding diagonal leaves from the complementary components to the stable lamination of a pseudoAnosov homeomorphism. We also give several new characterizations as well as a structure theorem for the dual lamination of T+(ϕ), in terms of Bestvina–Feighn–Handel’s ‘stable lamination ’ associated to ϕ. 1.
Currents twisting and nonsingular matrices
"... Abstract. We show that for k ≥ 3, given any matrix in GL(k, Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z k is the given matrix. 1. ..."
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Abstract. We show that for k ≥ 3, given any matrix in GL(k, Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z k is the given matrix. 1.
Random lengthspectrum rigidity for free groups
 Proceedings of AMS 140 (2012
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