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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Geometric intersection number and analogues of the curve complex for free groups

Ilya Kapovich and Martin Lustig

Geometry & Topology 13 (2009) 1805–1833

For the free group FN of finite rank N 2 we construct a canonical Bonahon-type, continuous and Out(FN)–invariant geometric intersection form

,: cv¯(FN) × Curr(FN) 0.

Here cv¯(FN) is the closure of unprojectivized Culler–Vogtmann Outer space cv(FN) in the equivariant Gromov–Hausdorff 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 –trees. The projectivization of cv¯(FN) 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.

free group, Outer space, geodesic current, curve complex
Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 57M99, 37B99, 37D99
Received: 26 August 2008
Accepted: 6 November 2008
Published: 20 March 2009
Proposed: Walter Neumann
Seconded: Martin Bridson, Benson Farb
Ilya Kapovich
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 West Green Street
Urbana, IL 61801
Martin Lustig
Mathématiques (LATP)
Université Paul Cézanne - Aix Marseille III
ave Escadrille Normandie-Niémen
13397 Marseille 20