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Volume entropy of a family of rank one, split-solvable Lie groups of Abelian type

Rene Garcia-Lara

Abstract

We study a family of metrics on Euclidean space that generalize the left-invariant metric of the SOL group and the metric of the logarithmic model of Hyperbolic space. Suppose G is a connected, simply-connected, Heintze group of Abelian type with diagonalizable derivation or the horospherical product of two such groups. In this scenario, G is isometric to Euclidean space with a metric of the type considered. We have derived a formula for the volume entropy of metrics in this family and used it to solve a conjecture related to a family of 3-manifolds that interpolates between the SOL group and hyperbolic space.

Volume entropy of a family of rank one, split-solvable Lie groups of Abelian type

Abstract

We study a family of metrics on Euclidean space that generalize the left-invariant metric of the SOL group and the metric of the logarithmic model of Hyperbolic space. Suppose G is a connected, simply-connected, Heintze group of Abelian type with diagonalizable derivation or the horospherical product of two such groups. In this scenario, G is isometric to Euclidean space with a metric of the type considered. We have derived a formula for the volume entropy of metrics in this family and used it to solve a conjecture related to a family of 3-manifolds that interpolates between the SOL group and hyperbolic space.

Paper Structure

This paper contains 10 sections, 25 theorems, 88 equations.

Table of Contents

  1. Statement of the problem and main results
  2. Geometry of the metric
  3. Logarithmic hyperbolic models of hyperbolic space
  4. Upper bounds for the growth of geodesic balls
  5. Lower bounds for the growth of geodesic balls
  6. Volume growth of geodesic balls: General case
  7. A lower bound for the volume of geodesic balls
  8. The area of a graph
  9. Computing the remaining of Cauchy-Schwartz
  10. Geodesic spheres are measurable

Key Result

Theorem 1

Let $a \in \mathbb{R}^{N}$, the volume entropy of $(\mathbb{R}^{N+1}, g_{a})$ is where the metric $g_{a}$ is given by Equation eq:the-horospheric-metric.

Theorems & Definitions (43)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • Proposition 8
  • ...and 33 more