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Margulis Multiverse: Infinitesimal Rigidity, Pressure Form and Convexity

Sourav Ghosh

Abstract

In this article we construct the pressure form on the moduli space of higher dimensional Margulis spacetimes without cusps and study its properties. We show that the Margulis spacetimes are infinitesimally determined by their marked Margulis invariant spectrums. We use it to show that the restrictions of the pressure form give Riemannian metrics on the constant entropy sections of the quotient moduli space. We also show that constant entropy sections of the moduli space with fixed linear part bound convex domains.

Margulis Multiverse: Infinitesimal Rigidity, Pressure Form and Convexity

Abstract

In this article we construct the pressure form on the moduli space of higher dimensional Margulis spacetimes without cusps and study its properties. We show that the Margulis spacetimes are infinitesimally determined by their marked Margulis invariant spectrums. We use it to show that the restrictions of the pressure form give Riemannian metrics on the constant entropy sections of the quotient moduli space. We also show that constant entropy sections of the moduli space with fixed linear part bound convex domains.

Paper Structure

This paper contains 13 sections, 36 theorems, 142 equations.

Table of Contents

  1. Preliminaries
  2. Thermodynamic formalism
  3. Anosov representations
  4. Margulis spacetimes and the Margulis invariant
  5. The moduli space and the Labourie--Margulis invariant
  6. Regularity of Labourie--Margulis invariants
  7. Infinitesimal rigidity
  8. Algebraicity of Margulis invariants
  9. Infinitesimal rigidity of Margulis invariants: fixed linear part
  10. Infinitesimal rigidity of Margulis invariants: general
  11. Pressure form and convexity
  12. Pressure form on the moduli space
  13. Margulis multiverses

Key Result

Proposition 1

Suppose $\mathsf{M}(\Gamma,\mathsf{G})$ is the space of all Margulis space times as defined above. Then $\mathsf{M}(\Gamma,\mathsf{G})$ is an open subset of the character variety and it is a fibered space. Moreover, each fiber is a disjoint union of two open convex cones in some vector space which d

Theorems & Definitions (77)

  • Proposition 1
  • Lemma 1
  • Proposition 2
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Theorem 2
  • Theorem 3
  • Remark 1.1
  • Remark 1.2
  • ...and 67 more