Orthospectra of geodesic laminations and dilogarithm identities on moduli space

Abstract

Given a measured lamination on a finite area hyperbolic surface we consider a natural measure M on the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation identities for the Rogers dilogarithm function on the moduli space of a surface. 32G15; 11M36

The Bridgeman–Kahn identity for hyperbolic manifolds with cusped boundary

“…We will start our proof by providing a full measure decomposition of the unit tangent bundle following the work of [BK10] and [Bri11]. We then proceed by calculating the volume of each piece in the decomposition.…”

“…In dimension 2, Bridgeman in [Bri11] gives an explicit formula for F 2 and also extends the identity to all finite-area orientable hyperbolic surfaces with totally geodesic boundary. His work yields the following beautiful identity: Let S be an oriented finite-area hyperbolic surface with nonempty totally geodesic boundary and m boundary cusps, then…”

Identities on hyperbolic manifolds

“…They also relate in this context to the McShane and Luo-Tan identities which we describe in the next section. The Bridgeman-Kahn identity in fact arose from a generalization of a previous paper of the first named author [11] which provided an explicit formula for the function F 2 (l) in Theorem B in terms of the Roger's dilogarithm. We have: where R is the Rogers dilogarithm function.…”

“…In the last couple of decades, several authors have discovered various remarkable and elegant identities on hyperbolic manifolds, including Basmajian [3], McShane [32,33], Bridgeman-Kahn [11,13] and Luo-Tan [28]. Some of these identities have been generalized and extended, with different and independent proofs given in some cases.…”

On volumes of truncated tetrahedra with constrained edge lengths

“…a geodesic arc intersecting the boundary of M orthogonally at its endpoints. Lengths of orthogeodesics define the orthospectrum, an interesting geometric object which has proved useful in the study of volumes of hyperbolic manifolds with geodesic boundary (see [Bas93,Bri11,BK10,Cal10,Cal11,BT14,BT16]). If M is compact, then the orthospectrum of M admits a positive minimum, say , hence no edge length of any truncated tetrahedron appearing in any decomposition of M can be smaller than ; moreover, the canonical Kojima decomposition of M contains a truncated polyhedron with one edge of length exactly equal to [Koj90,Koj92].…”