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

Kapovich, Ilya; Lustig, Martin

doi:10.2140/gt.2009.13.1805

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

Ilya Kapovich and Martin Lustig

Geometry & Topology 13 (2009) 1805–1833

DOI: 10.2140/gt.2009.13.1805

Abstract

For the free group $F_{N}$ of finite rank $N \geq 2$ we construct a canonical Bonahon-type, continuous and $Out (F_{N})$ –invariant geometric intersection form

〈, 〉 : \bar{cv} (F_{N}) \times Curr (F_{N}) \to ℝ_{\geq 0} .

Here $\bar{cv} (F_{N})$ 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 $\bar{cv} (F_{N})$ consists of all very small minimal isometric actions of $F_{N}$ on $ℝ$ –trees. The projectivization of $\bar{cv} (F_{N})$ 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.

Keywords

free group, Outer space, geodesic current, curve complex

Mathematical Subject Classification 2000

Primary: 20F65

Secondary: 57M99, 37B99, 37D99

References

Publication

Received: 26 August 2008

Accepted: 6 November 2008

Published: 20 March 2009

Proposed: Walter Neumann

Seconded: Martin Bridson, Benson Farb

Authors

Ilya Kapovich
Department of Mathematics University of Illinois at Urbana-Champaign 1409 West Green Street Urbana, IL 61801 USA
http://www.math.uiuc.edu/~kapovich/
Martin Lustig
Mathématiques (LATP) Université Paul Cézanne - Aix Marseille III ave Escadrille Normandie-Niémen 13397 Marseille 20 France

Volume 13, issue 3 (2009)