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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 ..."
Abstract

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.
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)
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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].