Geometry & Topology

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

Ilya Kapovich and Martin Lustig

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Abstract

For the free group FN of finite rank N2 we construct a canonical Bonahon-type, continuous and Out(FN)–invariant geometric intersection form

, : cv ¯ ( F N ) × Curr ( F N ) 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.

Article information

Source
Geom. Topol., Volume 13, Number 3 (2009), 1805-1833.

Dates
Received: 26 August 2008
Accepted: 6 November 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800259

Digital Object Identifier
doi:10.2140/gt.2009.13.1805

Mathematical Reviews number (MathSciNet)
MR2496058

Zentralblatt MATH identifier
1194.20046

Subjects
Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 57M99: None of the above, but in this section 37B99: None of the above, but in this section 37D99: None of the above, but in this section

Keywords
free group Outer space geodesic current curve complex

Citation

Kapovich, Ilya; Lustig, Martin. Geometric intersection number and analogues of the curve complex for free groups. Geom. Topol. 13 (2009), no. 3, 1805--1833. doi:10.2140/gt.2009.13.1805. https://projecteuclid.org/euclid.gt/1513800259


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