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 have \[ \sum_{\gamma} \frac{1}{e^{\ell(\gamma)}+1}={1/2} \] where $\gamma$ varies over the homotopy classes of essential simple closed curves and $\ell(\gamma)$ is the length of the geodesic representative of $\gamma$. We prove that there is no reasonable analogue of McShane's identity for the Culler-Vogtmann outer space of a free group.