The Bridgeman-Kahn identity for hyperbolic manifolds with cusped boundary

TL;DR

This paper extends the Bridgeman-Kahn identity to finite-volume hyperbolic n-manifolds with cusped boundary, incorporating boundary cusp invariants into the volume expression, thus broadening its applicability.

Contribution

The authors generalize the Bridgeman-Kahn identity to include manifolds with cusped boundary, adding terms for boundary cusp invariants.

Findings

Extended identity includes boundary cusp invariants.

Expressed volume as sum over orthospectrum and cusp terms.

Applicable to all finite-volume orientable hyperbolic manifolds.

Abstract

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.