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Volume, entropy, and diameter in ${\rm SO}(p,q+1)$-higher Teichmüller spaces

Filippo Mazzoli, Gabriele Viaggi

Abstract

We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations $ρ: Γ\to \mathrm{SO}(p,q+1)$ of closed $p$-manifold groups. In particular: We provide a uniform lower bound of the product entropy times volume that depends only on the geometry of the abstract group $Γ$. We prove that the entropy is bounded from above by $p-1$ with equality if and only if $ρ$ is conjugate to a representation inside ${\rm S}({\rm O}(p,1)\times{\rm O}(q))$, which answers affirmatively to a question of Glorieux and Monclair. Lastly, we prove finiteness and compactness results for groups admitting convex cocompact representations with bounded diameter.

Volume, entropy, and diameter in ${\rm SO}(p,q+1)$-higher Teichmüller spaces

Abstract

We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations of closed -manifold groups. In particular: We provide a uniform lower bound of the product entropy times volume that depends only on the geometry of the abstract group . We prove that the entropy is bounded from above by with equality if and only if is conjugate to a representation inside , which answers affirmatively to a question of Glorieux and Monclair. Lastly, we prove finiteness and compactness results for groups admitting convex cocompact representations with bounded diameter.
Paper Structure (17 sections, 24 theorems, 133 equations)

This paper contains 17 sections, 24 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Spatial distances
  3. Ricci curvature of spacelike submanifolds
  4. Fermi charts
  5. Distance and Busemann functions
  6. A compactness criterion
  7. Lower bound for the gradient
  8. Upper bound for the gradient
  9. Proof of Theorem \ref{['thm:distance comparison']}
  10. A fundamental domain
  11. Volume, entropy, and topology
  12. Proof of Proposition \ref{['pro:volume']}
  13. Entropy rigidity
  14. Finiteness and compactness
  15. Proof of Theorem \ref{['thm:main4']}
  16. ...and 2 more sections

Key Result

Theorem 1

Let $M$ be a complete, spacelike, maximal $p$-submanifold of $\mathbb{H}^{p,q}$. Then its Ricci curvature and second fundamental form $\mathbb{I}$ satisfy where $g_M$ denotes the induced metric of $M$.

Theorems & Definitions (56)

  • Theorem : Ishihara I*Proposition 2.1, Theorem 1.2
  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Lemma 2.1
  • ...and 46 more