Results 1  10
of
40
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. ..."
Abstract

Cited by 27 (13 self)
 Add to MetaCart
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
On hyperbolicity of free splitting and free factor complexes
 Groups, Geom. Dynam
"... Abstract. We show how to derive hyperbolicity of the free factor complex of FN from the HandelMosher proof of hyperbolicity of the free splitting complex of FN, thus obtaining an alternative proof of a theorem of BestvinaFeighn. We also show that under the natural map τ from the free splitting com ..."
Abstract

Cited by 21 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We show how to derive hyperbolicity of the free factor complex of FN from the HandelMosher proof of hyperbolicity of the free splitting complex of FN, thus obtaining an alternative proof of a theorem of BestvinaFeighn. We also show that under the natural map τ from the free splitting complex to free factor complex, a geodesic [x, y] maps to a path that is uniformly Hausdorffclose to a geodesic [τ(x), τ(y)]. 1.
Growth of intersection numbers for free group automorphisms
 J. Topol
"... Abstract. For a fully irreducible automorphism φ of the free group Fk we compute the asymptotics of the intersection number n↦ → i(T, T ′ φ n) for trees T, T ′ in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T ′ φ n for n large. ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
(Show Context)
Abstract. For a fully irreducible automorphism φ of the free group Fk we compute the asymptotics of the intersection number n↦ → i(T, T ′ φ n) for trees T, T ′ in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T ′ φ n for n large.
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 ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
(Show Context)
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. ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
(Show Context)
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].
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 ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
(Show Context)
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.
The complex of partial bases for Fn and finite generation of the Torelli subgroup of Aut(Fn)
, 2012
"... We study the complex of partial bases of a free group, which is an analogue for Aut(Fn) of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of Aut(Fn) is highly connected. Using these result ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
We study the complex of partial bases of a free group, which is an analogue for Aut(Fn) of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of Aut(Fn) is highly connected. Using these results, we give a new, topological proof of a theorem of Magnus that asserts that the Torelli subgroup of Aut(Fn) is finitely generated.