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Volumes of hyperbolic manifolds and mixed Tate motives

Alexander Goncharov

TL;DR

The paper connects the geometry of odd-dimensional hyperbolic manifolds to algebraic K-theory via the Borel regulator, showing volumes correspond to regulator evaluations of elements in $K_{2n-1}({\bar{\mathbb{Q}}})\otimes \mathbb{Q}$. It develops two invariant constructions that respect scissor-congruence, extends the Dehn-invariant framework to higher dimensions, and provides a motivic interpretation through mixed Tate motives, linking volume data to Beilinson regulators. By constructing a Hopf-algebraic model $S(\mathbb{C})_{\bullet}$ and embedding it into the category of mixed Tate motives, the work derives conjectures that tie geometric invariants to graded pieces of $K$-theory, with concrete results in low dimensions (notably $n=2$) and clear pathways for polylogarithm-based realizations. The approach merges hyperbolic geometry, continued Dehn-complex analysis, and motivic homotopy theory to illuminate the transcendental nature of volumes and their regulator images, offering a conceptual framework for future computations and for understanding scissor-congruence beyond three dimensions.

Abstract

Two different constructions of an invariant of an odd dimensional hyperbolic manifold in the K-group $K_{2n-1}(\bar \Bbb Q)\otimes \Bbb Q$ are given. The volume of the manifold is equal to the value of the Borel regulator on that element. The scissor congruence groups in non euclidian geometries are studied and their relationship with algebraic K-theory of the field of complex numbers is discussed.

Volumes of hyperbolic manifolds and mixed Tate motives

TL;DR

The paper connects the geometry of odd-dimensional hyperbolic manifolds to algebraic K-theory via the Borel regulator, showing volumes correspond to regulator evaluations of elements in . It develops two invariant constructions that respect scissor-congruence, extends the Dehn-invariant framework to higher dimensions, and provides a motivic interpretation through mixed Tate motives, linking volume data to Beilinson regulators. By constructing a Hopf-algebraic model and embedding it into the category of mixed Tate motives, the work derives conjectures that tie geometric invariants to graded pieces of -theory, with concrete results in low dimensions (notably ) and clear pathways for polylogarithm-based realizations. The approach merges hyperbolic geometry, continued Dehn-complex analysis, and motivic homotopy theory to illuminate the transcendental nature of volumes and their regulator images, offering a conceptual framework for future computations and for understanding scissor-congruence beyond three dimensions.

Abstract

Two different constructions of an invariant of an odd dimensional hyperbolic manifold in the K-group are given. The volume of the manifold is equal to the value of the Borel regulator on that element. The scissor congruence groups in non euclidian geometries are studied and their relationship with algebraic K-theory of the field of complex numbers is discussed.

Paper Structure

This paper contains 5 sections, 14 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Dehn complexes in classical geometries and their homology
  3. Volumes of hyperbolic $n$-manifolds and continuous cohomology of $SO(n,1)$
  4. Noneuclidean polyhedrons and mixed Tate motives
  5. Proof of the theorem 2.5

Key Result

Theorem 1.1

Any $(2m-1)$-dimensional hyperbolic manifold of finite volume $M^{2m-1}$ defines naturally an element $\gamma(M^{2m-1}) \in K_{2m-1}({\bar{{\Bbb Q}}}){\otimes}{\Bbb Q}$ such that ${\hbox{vol}}(M^{2m-1}) = r_{m} (\gamma(M^{2m-1}))$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Conjecture 2.4
  • Theorem 2.5
  • Conjecture 2.6
  • ...and 10 more