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Collapsing and group growth as obstructions to Einstein metrics on some smooth 4-manifolds

Haydeé Contreras Peruyero, Pablo Suárez-Serrato

Abstract

We show that a combination of collapsing and excessive growth from the fundamental group impedes the existence of Einstein metrics on several families of smooth four-manifolds. These include infrasolvmanifolds whose fundamental group is not virtually nilpotent, most elliptic surfaces of zero Euler characteristic, geometrizable manifolds with hyperbolic factor geometries in their geometric decomposition, and higher graph four-manifolds without purely negatively curved pieces.

Collapsing and group growth as obstructions to Einstein metrics on some smooth 4-manifolds

Abstract

We show that a combination of collapsing and excessive growth from the fundamental group impedes the existence of Einstein metrics on several families of smooth four-manifolds. These include infrasolvmanifolds whose fundamental group is not virtually nilpotent, most elliptic surfaces of zero Euler characteristic, geometrizable manifolds with hyperbolic factor geometries in their geometric decomposition, and higher graph four-manifolds without purely negatively curved pieces.

Paper Structure

This paper contains 8 sections, 14 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Ancillaries & Preliminaries
  3. Collapsing
  4. Simplicial Volume
  5. Group growth
  6. Growth of generating sets
  7. Growth types of groups
  8. Proofs

Key Result

Theorem 1

Let $M$ be a closed orientable smooth $4$--manifold modeled on one of the Thurston geometries $\mathbb{S}^{3}\times \mathbb{E},\, \mathbb{N}il^{3}\times \mathbb{E},\, \mathbb{N}il^{4}, \, \mathbb{H}^{3}\times \mathbb{E},\, \widetilde{\rm SL}_{2}\times \mathbb{E},\, \mathbb{H}^{2}\times\mathbb{E}^{2}

Theorems & Definitions (21)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Definition 1
  • Theorem 4
  • Example 1
  • Theorem 5
  • Definition 2: CS19
  • Example 2
  • Example 3
  • ...and 11 more