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Growth in the universal cover under large simplicial volume

Hannah Alpert

Abstract

Consider a closed manifold $M$ with two Riemannian metrics: one hyperbolic metric, and one other metric $g$. What hypotheses on $g$ guarantee that for a given radius $r$, there are balls of radius $r$ in the universal cover of $(M, g)$ with greather-than-hyperbolic volumes? We show that this conclusion holds for all $r \geq 1$ if $(\mathrm{Vol} (M, g))^2$ is less than a small constant times the hyperbolic volume of $M$. This strengthens a theorem of Sabourau and is partial progress toward a conjecture of Guth.

Growth in the universal cover under large simplicial volume

Abstract

Consider a closed manifold with two Riemannian metrics: one hyperbolic metric, and one other metric . What hypotheses on guarantee that for a given radius , there are balls of radius in the universal cover of with greather-than-hyperbolic volumes? We show that this conclusion holds for all if is less than a small constant times the hyperbolic volume of . This strengthens a theorem of Sabourau and is partial progress toward a conjecture of Guth.
Paper Structure (6 sections, 10 theorems, 31 equations)

This paper contains 6 sections, 10 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Relationship with Sabourau's proof
  3. Amenable Reduction Lemma
  4. Separating filtrations
  5. Main proof
  6. Open questions

Key Result

Theorem 1

For every $n \geq 2$, there exists $\delta_n > 0$ such that if $M$ is a closed, oriented, connected $n$-dimensional manifold admitting a hyperbolic metric, and $g$ is another Riemannian metric on $M$ with $\frac{(\mathop{\mathrm{Vol}}\nolimits (M, g))^2}{\Vert M \Vert_{\Delta}} < \delta_n$, then for where $\widetilde{M}$ denotes the universal cover of $(M, g)$, and $\mathbb{H}^n$ denotes the hyper

Theorems & Definitions (14)

  • Theorem 1: Main theorem
  • Theorem 2: sabourau22
  • Theorem 3: Theorem 2 of guth11
  • Conjecture 4: guth11
  • Theorem 5: dey19
  • Lemma 6: sabourau22
  • proof : Proof summary
  • Theorem 7: Amenable Reduction Lemma
  • Lemma 8: Lemma 4 of alpert22
  • Lemma 9: Lemma 7 of alpert22
  • ...and 4 more