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Geometric Intersection Number and analogues of the Curve Complex for free groups. preprint arXiv:math/0711.3806
"... Abstract. 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 GromovHa ..."
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Cited by 42 (15 self)
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Abstract. 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. 1.
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).
The PattersonSullivan embedding and minimal volume entropy for Outer space
 Geom. Funct. Anal. (GAFA
"... Abstract. Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the PattersonSullivanBowenMargulis map from the CullerVogtmann outer space CV (Fk) into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding and ..."
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Cited by 21 (14 self)
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Abstract. Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the PattersonSullivanBowenMargulis map from the CullerVogtmann outer space CV (Fk) into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding and thus obtain a new compactification of the outer space. We also prove that for every k ≥ 2 the minimum of the volume entropy of the universal covers of finite connected volumeone metric graphs with fundamental group of rank k and without degreeone vertices is equal to (3k − 3)log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs. 1.
Translation equivalent elements in free groups
 J. Group Theory
"... Abstract. Let Fn be a free group of rank n ≥ 2. Two elements g, h in Fn are said to be translation equivalent in Fn if the cyclic length of φ(g) equals the cyclic length of φ(h) for every automorphism φ of Fn. Let F(a, b) be the free group generated by {a, b} and let w(a, b) be an arbitrary word in ..."
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Cited by 8 (2 self)
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Abstract. Let Fn be a free group of rank n ≥ 2. Two elements g, h in Fn are said to be translation equivalent in Fn if the cyclic length of φ(g) equals the cyclic length of φ(h) for every automorphism φ of Fn. Let F(a, b) be the free group generated by {a, b} and let w(a, b) be an arbitrary word in F(a, b). We prove that w(g, h) and w(h, g) are translation equivalent in Fn whenever g, h ∈ Fn are translation equivalent in Fn, which hereby gives an affirmative solution to problem F38b in the online version
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.
AN ALGORITHM THAT DECIDES TRANSLATION EQUIVALENCE IN A FREE GROUP OF RANK TWO
, 2006
"... Abstract. Let F2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for given two elements u, v of F2, u and v are translation equivalent in F2, that is, whether or not u and v have the property that the cyclic length of φ(u) equals the cyclic length of φ(v) ..."
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Cited by 4 (1 self)
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Abstract. Let F2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for given two elements u, v of F2, u and v are translation equivalent in F2, that is, whether or not u and v have the property that the cyclic length of φ(u) equals the cyclic length of φ(v) for every automorphism φ of F2. This gives an affirmative solution to problem F38a in the online version (http://www.grouptheory.info) of [1] for the case of F2. 1.
Domains of proper discontinuity on the boundary of Outer space
"... Abstract. Motivated by the work of McCarthy and Papadopoulos for subgroups of mapping class groups, we construct domains of proper discontinuity in the compactified Outer space and in the projectivized space of geodesic currents for any “sufficiently large ” subgroup of Out(FN) (that is, a subgroup ..."
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Cited by 3 (2 self)
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Abstract. Motivated by the work of McCarthy and Papadopoulos for subgroups of mapping class groups, we construct domains of proper discontinuity in the compactified Outer space and in the projectivized space of geodesic currents for any “sufficiently large ” subgroup of Out(FN) (that is, a subgroup containing a hyperbolic iwip). As a corollary we prove that for N ≥ 3 the action of Out(FN) on the subset of PCurr(FN) consisting of all projectivized currents with full support is properly discontinuous. 1.
ON SEVERAL PROBLEMS ABOUT AUTOMORPHISMS OF THE FREE GROUP OF RANK TWO
, 802
"... Abstract. Let Fn be a free group of rank n. In this paper we discuss three algorithmic problems related to automorphisms of F2. A word u of Fn is called positive if u does not have negative exponents. A word u in Fn is called potentially positive if φ(u) is positive for some automorphism φ of Fn. We ..."
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Cited by 2 (0 self)
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Abstract. Let Fn be a free group of rank n. In this paper we discuss three algorithmic problems related to automorphisms of F2. A word u of Fn is called positive if u does not have negative exponents. A word u in Fn is called potentially positive if φ(u) is positive for some automorphism φ of Fn. We prove that there is an algorithm to decide whether or not a given word in F2 is potentially positive, which gives an affirmative solution to problem F34a in [1] for the case of F2. Two elements u and v in Fn are said to be boundedly translation equivalent if the ratio of the cyclic lengths of φ(u) and φ(v) is bounded away from 0 and from ∞ for every automorphism φ of Fn. We provide an algorithm to determine whether or not two given elements of F2 are boundedly translation equivalent, thus answering question F38c in the online version of [1] for the case of F2. We further prove that there exists an algorithm to decide whether or not a given finitely generated subgroup of F2 is the fixed point group of some automorphism of F2, which settles problem F1b in [1] in the affirmative for the case of F2. 1.