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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.
Intersection form, laminations and currents on free groups
, 2009
"... Let F be a free group of rank N ≥ 2, let µ be a geodesic current on F and let T be an Rtree 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. ..."
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Cited by 27 (13 self)
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Let F be a free group of rank N ≥ 2, let µ be a geodesic current on F and let T be an Rtree 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 a toroidal 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 some of the translation length functions of any two ”sufficiently transverse” very small Ftrees 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 actions of Out(Fk) on the boundary of outer space and on the space of currents: minimal sets and equivariant incompatibility
, 2006
"... We prove that for k ≥ 5 there does not exist a continuous map ∂CV (Fk) → PCurr(Fk) that is either Out(Fk)equivariant or Out(Fk)antiequivariant. Here ∂CV (Fk) is the “lengthfunction” boundary of CullerVogtmann’s Outer space CV (Fk), and PCurr(Fk) is the space of projectivized geodesic currents ..."
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Cited by 25 (14 self)
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We prove that for k ≥ 5 there does not exist a continuous map ∂CV (Fk) → PCurr(Fk) that is either Out(Fk)equivariant or Out(Fk)antiequivariant. Here ∂CV (Fk) is the “lengthfunction” boundary of CullerVogtmann’s Outer space CV (Fk), and PCurr(Fk) is the space of projectivized geodesic currents for Fk. We also prove that, if k ≥ 3, for the action of Out(Fk) on PCurr(Fk) and for the diagonal action of Out(Fk) on the product space ∂CV (Fk)× PCurr(Fk) there exist unique nonempty minimal closed Out(Fk)invariant sets. Our results imply that for k ≥ 3 any continuous Out(Fk)equivariant embedding of CV (Fk) into PCurr(Fk) (such as the PattersonSullivan embedding) produces a new compactification of Outer space, different from the usual “lengthfunction” compactification CV (Fk) = CV (Fk) ∪ ∂CV (Fk).
Minimal Volume Entropy on Graphs
, 2005
"... Among the normalized metrics on a graph, we show the existence and the uniqueness of an entropyminimizing metric, and give explicit formulas for the minimal volume entropy and the metric realizing it. Parmi les distances normalisées sur un graphe, nous montrons l’existence et l’unicité d’une distan ..."
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Cited by 12 (1 self)
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Among the normalized metrics on a graph, we show the existence and the uniqueness of an entropyminimizing metric, and give explicit formulas for the minimal volume entropy and the metric realizing it. Parmi les distances normalisées sur un graphe, nous montrons l’existence et l’unicité d’une distance qui minimise l’entropie, et nous donnons des formules explicites pour l’entropie volumique minimale et la distance qui la réalise. 1.Introduction Let (X, g) be a compact connected Riemannian manifold of nonpositive curvature. It was shown by A. Manning [Man] that the topological entropy htop(g) of the geodesic flow is equal to the volume entropy hvol(g) of the manifold 1 hvol(g) = lim log(vol(B(x, r))),
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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Cited by 10 (5 self)
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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.
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].
CURRENTS ON FREE GROUPS
, 2005
"... We study the properties of geodesic currents on free groups, particularly the “intersection form” that is similar to Bonahon’s notion of the intersection number between geodesic currents on hyperbolic surfaces. ..."
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Cited by 8 (2 self)
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We study the properties of geodesic currents on free groups, particularly the “intersection form” that is similar to Bonahon’s notion of the intersection number between geodesic currents on hyperbolic surfaces.
Random lengthspectrum rigidity for free groups
 Proceedings of AMS 140 (2012
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Geometric entropy of geodesic currents on free groups
 Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, Contemporary Mathematics series, American Mathematical Society
"... Abstract. A geodesic current on a free group F is an Finvariant measure on the set ∂ 2 F of pairs of distinct points of ∂F. The space of geodesic currents on F is a natural companion of CullerVogtmann’s Outer space cv(F) and studying them together yields new information about both spaces as well a ..."
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Cited by 4 (3 self)
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Abstract. A geodesic current on a free group F is an Finvariant measure on the set ∂ 2 F of pairs of distinct points of ∂F. The space of geodesic currents on F is a natural companion of CullerVogtmann’s Outer space cv(F) and studying them together yields new information about both spaces as well as about the group Out(F). The main aim of this paper is to introduce and study the notion of geometric entropy hT(µ) of a geodesic current µ with respect to a point T of cv(F), which can be viewed as a length function on F. The geometric entropy is defined as the slowest rate of exponential decay of µmeasures of biinfinite cylinders in F, as the Tlength of the word defining such a cylinder goes to infinity. We obtain an explicit formula for h T ′(µT), where T, T ′ are arbitrary points in cv(F) and where µT denotes a PattersonSullivan current corresponding to T. It involves the volume entropy h(T) and the extremal distortion of distances in T with respect to distances in T ′. It follows that, given T in the projectivized outer space CV (F), h T ′(µT) as function of T ′ ∈ CV (F) achieves a strict global maximum at T ′ = T. We also show that for any T ∈ cv(F) and any geodesic current µ on F, hT (µ) ≤ h(T), where the equality is realized when µ = µT. For points T ∈ cv(F) with simplicial metric (where all edges have length one), we relate the geometric entropy of a current and the measuretheoretic entropy. 1.