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The Bridgeman-Kahn identity for hyperbolic manifolds with cusped boundary

Nicholas G. Vlamis, Andrew Yarmola

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.

The Bridgeman-Kahn identity for hyperbolic manifolds with cusped boundary

Abstract

In this note, we extend the Bridgeman-Kahn identity to all finite-volume orientable hyperbolic -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.

Paper Structure

This paper contains 9 sections, 11 theorems, 58 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background on hyperbolic manifolds
  3. Cusps
  4. Unit tangent bundle
  5. An Apollonian manifold
  6. Identity for manifolds with cusped boundary
  7. Decomposition of the Unit Tangent Bundle
  8. Computing $\mathop{\mathrm{Vol}}\nolimits(V_\mathfrak{c})$
  9. Integration

Key Result

Theorem 1.1

For $n \geq 3$ and $M$ an oriented finite-volume hyperbolic $n$-manifold with nonempty totally geodesic boundary, let $\mathfrak{B}$ be the set of boundary cusps of $M$ and let $|\mathcal{O}(M)|$ be the orthospectrum. For every $\mathfrak{c} \in \mathfrak{B}$, let $B_\mathfrak{c}$ be an embedded hor where $F_n$ is the $n^{th}$-Bridgeman-Kahn function, $\mathop{\mathrm{Vol}}\nolimits$ is the hyperb

Figures (7)

  • Figure 1: Whitehead link
  • Figure 2: A piece of the Apollonian strip in $\partial \mathbb U^3$
  • Figure 3: $\widetilde{B}_\mathfrak{r} \cup \widetilde{B}_\mathfrak{b}$ wth a red (light) cusp at infinity.
  • Figure 4: $\widetilde{B}_\mathfrak{r} \cup \widetilde{B}_\mathfrak{b}$ with a blue (dark) cusp at infinity.
  • Figure 5: To compute $\mathop{\mathrm{Vol}}\nolimits(V_\mathfrak{c})$, we must find the volume of all vectors $v \in T_1V$ for which the corresponding complete geodesic emanates from $D$ and terminates in $U_+$.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.1
  • Theorem 4.1
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 9 more