Geometriae Dedicata

, Volume 194, Issue 1, pp 81–97 | Cite as

The Bridgeman–Kahn identity for hyperbolic manifolds with cusped boundary

  • Nicholas G. Vlamis
  • Andrew YarmolaEmail author
Original Paper


In this note, we extend the Bridgeman–Kahn identity to all finite-volume orientable hyperbolic n-manifolds with totally geodesic boundary. In the compact case, Bridgeman and Kahn are able to express the manifold’s volume as the sum of a function over only the orthospectrum. For manifolds with non-compact boundary, our extension adds terms corresponding to intrinsic invariants of boundary cusps.


Hyperbolic manifold Geodesic boundary Orthospectrum Geometric identities 

Mathematics Subject Classification

57M50 30F40 32Q45 



The authors thank Martin Bridgeman for suggesting the problem.


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Mathematics Research UnitMaison du NombreEsch-sur-AlzetteLuxembourg

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