Geometric intersection number and analogues of the curve complex for free groups

Abstract

1805Geometric intersection number and analogues of the curve complex for free groups ILYA KAPOVICH MARTIN LUSTIGFor the free group F N of finite rank N 2 we construct a canonical Bonahon-type, continuous and Out.F N /-invariant geometric intersection formHere cv.F N / is the closure of unprojectivized Culler-Vogtmann Outer space cv.F N / in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that cv.F N / consists of all very small minimal isometric actions of F N on R-trees. The projectivization of cv.F N / provides a free group analogue of Thurston's compactification of 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

“…The intersection form was introduced in [Ka3,4], [Lu] for free simplicial actions of F , that is for the non-projectivized outer space cv (F ). In a recent paper [KaLu2] we proved that the intersection form extends continuously to the closure cv (F ) of cv (F ) consisting of all minimal very small isometric actions of F on R-trees. Note that the projectivization Pcv(F ) of cv (F ) is exactly the compactification CV (F ) of CV (F ).…”

“…For T ∈ cv (F ) and for µ ∈ Curr(F ) we will also call T, µ and the geometric intersection number of T and µ. In general, if T ∈ cv (F ) and if µ ∈ Curr(F ) is approximated by rational currents as µ = lim i→∞ λ i η g i , where g i ∈ F , λ i ≥ 0, the geometric intersection number T, µ can be computed as T, µ = lim i→∞ λ i g i T ( ‡) Ursula Hamenstädt [H] recently used our result from [KaLu2] about the continuous extension of the intersection form to cv (F ) as a key ingredient to prove that any non-elementary subgroup of Out (F ), where N ≥ 3, has infinite-dimensional second bounded cohomology group (infinite-dimensional space of quasi-morphisms). This in turn has an application to proving that any homomorphism from any lattice in a higher-rank semi-simple Lie group to Out (F ), where N ≥ 3, has finite image.…”

“…Very recently Bestvina and Feighn [BeF2] used [KaLu2] to show that for any finite collection φ 1 , . .…”

“…One of the main motivations and prospective uses for Theorem 1.1 is to analyze the intersection graph, introduced by the authors in [KaLu2] in order to study various free group analogues of the curve complex. The intersection graph I(F ) is a bipartite graph with the vertex set Pcv(F ) P Curr(F ) where [T ] ∈ Pcv(F ) and [µ] ∈ P Curr(F ) are adjacent in I(F ) if and only if T, µ = 0.…”

“…Taking a dual point of view, one can think of an essential simple closed curve on a surface as a splitting of the surface group over Z, where two curves are adjacent in the curve complex if and only if the corresponding Z-splittings have a common refinement. This leads to the notion [KaLu2] of a cut graph for F whose vertices are nontrivial splittings of F as the fundamental group of a graph of groups with a single edge and the trivial edge group, and where adjacency again corresponds to having a common refinement. A variation of this construction would declare two splittings to be adjacent if there exists a nontrivial element in F which is elliptic with respect to both of them.…”