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

2006/02/14 by Kapovich, Ilya, Rivin, Igor
#20F65 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.math/0602291

Abstract

A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we have ∑_γ \frac1e^ℓ(γ)+1=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.

Related

• McShane's identity, using elliptic elements
• McShane's identity for classical Schottky Groups
• A McShane-type identity for closed surfaces
• Lengths of geodesics on non-orientable hyperbolic surfaces
• Geodesic Length Functions and Teichmüller Spaces