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

A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we have 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 McShaneʹs identity for the Culler–Vogtmann outer space of a free group.