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

Ilya Kapovich·Igor Rivin
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Abstract

A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we have \[ \sum_γ \frac{1}{e^{\ell(γ)}+1}={1/2} \] where $γ$ varies over the homotopy classes of essential simple closed curves and $\ell(γ)$ 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.

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Ilya Kapovich, Igor Rivin. 2006-02-14. On the absence of McShane-type identities for the outer space. https://arxiv.org/abs/math/0602291

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