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Volume growth of horospheres in diagonalizable Heintze groups

Gilles Courtois, Pablo Lessa, Emiliano Sequeira

TL;DR

This work analyzes the intrinsic geometry of horospheres in real diagonal Heintze groups $G_A$ by constructing an approximate horosphere $\\mathcal{H}$ and proving it is quasi-isometric to actual horospheres. The main technical device is a controlled comparison via a distance surrogate $\\rho$ and a convex, piecewise-smooth model of a horosphere, enabling explicit volume-growth computations. The authors show there are exactly two isometry (and hence two quasi-isometry) classes of horospheres when $\\lambda_1 < \\lambda_d$: one Euclidean class with growth like $r^d$, and a non-Euclidean class with growth order $r^k$ where $k = \\frac{\\lambda_1+\\cdots+\\lambda_d}{\\lambda_1}$. This dichotomy ties to boundary geometry, including the conformal dimension of the boundary and the $L^p$-cohomology exponent, and demonstrates rigidity of horosphere geometry under quasi-isometries in these spaces. The methodology combines geometric analysis on Lie groups, convex-geometry projections, and Coulhon–Saloff-Coste-type isometry-at-infinity results to relate local and asymptotic volume growth.

Abstract

We study the volume growth of horospheres in a Heintze group of the form R ___ A R d with A a diagonal derivation. We conclude that the isometry and quasi-isometry classes of horospheres (with their intrinsic geometry) coincide. Furthermore, if A is not a scalar multiple of the identity, then there are exactly two such classes, characterized by their volume growth, which we calculate explicitly.

Volume growth of horospheres in diagonalizable Heintze groups

TL;DR

This work analyzes the intrinsic geometry of horospheres in real diagonal Heintze groups by constructing an approximate horosphere and proving it is quasi-isometric to actual horospheres. The main technical device is a controlled comparison via a distance surrogate and a convex, piecewise-smooth model of a horosphere, enabling explicit volume-growth computations. The authors show there are exactly two isometry (and hence two quasi-isometry) classes of horospheres when : one Euclidean class with growth like , and a non-Euclidean class with growth order where . This dichotomy ties to boundary geometry, including the conformal dimension of the boundary and the -cohomology exponent, and demonstrates rigidity of horosphere geometry under quasi-isometries in these spaces. The methodology combines geometric analysis on Lie groups, convex-geometry projections, and Coulhon–Saloff-Coste-type isometry-at-infinity results to relate local and asymptotic volume growth.

Abstract

We study the volume growth of horospheres in a Heintze group of the form R ___ A R d with A a diagonal derivation. We conclude that the isometry and quasi-isometry classes of horospheres (with their intrinsic geometry) coincide. Furthermore, if A is not a scalar multiple of the identity, then there are exactly two such classes, characterized by their volume growth, which we calculate explicitly.

Paper Structure

This paper contains 9 sections, 24 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Geometric preliminaries
  3. Approximating horosphere
  4. Approximate distance
  5. Approximate horosphere
  6. Volume growth of the approximate horosphere
  7. The approximate horosphere has controlled volume
  8. Quasi-isometry
  9. Isometry at infinity

Key Result

Theorem 1

Let $G_A$ be a real diagonal Heintze group. Assume $\lambda_1 < \lambda_d$, then there exist exactly two isometry classes of horospheres in $G_A$. One class consists of horospheres isometric to the Euclidean space $\mathbb R^d$, and the other of horospheres which have volume growth of order $r^{k}$

Theorems & Definitions (48)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • Lemma 1
  • proof
  • Corollary 2
  • ...and 38 more