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.
Categories: math.GR math.GT