On the absence of McShane-type identities for the outer space

Ilya Kapovich; Igor Rivin

doi:10.1016/j.jalgebra.2008.08.024

On the absence of McShane-type identities for the outer space

Ilya Kapovich, Igor Rivin

Research output: Contribution to journal › Article › peer-review

Abstract

A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we haveunder(∑, γ) frac(1, e^{ℓ (γ)} + 1) = frac(1, 2) where γ varies over the homotopy classes of essential simple closed curves and ℓ (γ) is the length of the geodesic representative of γ. We prove that there is no reasonable analogue of McShane's identity for the Culler-Vogtmann outer space of a free group.

Original language	English (US)
Pages (from-to)	3659-3670
Number of pages	12
Journal	Journal of Algebra
Volume	320
Issue number	10
DOIs	https://doi.org/10.1016/j.jalgebra.2008.08.024
State	Published - Nov 15 2008
Externally published	Yes

Keywords

Free groups
Outer space

ASJC Scopus subject areas

Algebra and Number Theory

Online availability

10.1016/j.jalgebra.2008.08.024

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title = "On the absence of McShane-type identities for the outer space",

abstract = "A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we haveunder(∑, γ) frac(1, eℓ (γ) + 1) = frac(1, 2) where γ varies over the homotopy classes of essential simple closed curves and ℓ (γ) is the length of the geodesic representative of γ. We prove that there is no reasonable analogue of McShane's identity for the Culler-Vogtmann outer space of a free group.",

keywords = "Free groups, Outer space",

author = "Ilya Kapovich and Igor Rivin",

note = "Funding Information: E-mail addresses: kapovich@math.uiuc.edu (I. Kapovich), rivin@euclid.math.temple.edu (I. Rivin). URL: http://www.math.uiuc.edu/~kapovich/ (I. Kapovich). 1 The author was supported by the NSF grants DMS#0404991, DMS#0603921 and by the Humboldt Foundation Fellowship.",

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doi = "10.1016/j.jalgebra.2008.08.024",

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N2 - A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we haveunder(∑, γ) frac(1, eℓ (γ) + 1) = frac(1, 2) where γ varies over the homotopy classes of essential simple closed curves and ℓ (γ) is the length of the geodesic representative of γ. We prove that there is no reasonable analogue of McShane's identity for the Culler-Vogtmann outer space of a free group.

AB - A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we haveunder(∑, γ) frac(1, eℓ (γ) + 1) = frac(1, 2) where γ varies over the homotopy classes of essential simple closed curves and ℓ (γ) is the length of the geodesic representative of γ. We prove that there is no reasonable analogue of McShane's identity for the Culler-Vogtmann outer space of a free group.

KW - Free groups

KW - Outer space

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On the absence of McShane-type identities for the outer space

Abstract

Keywords

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