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Simplicial volume and 0-strata of separating filtrations

Hannah Alpert

Abstract

We use Papasoglu's method of area-minimizing separating sets to give an alternative proof, and explicit constants, for the following theorem of Guth and Braun--Sauer: If $M$ is a closed, oriented, $n$-dimensional manifold, with a Riemannian metric such that every ball of radius $1$ in the universal cover of $M$ has volume at most $V_1$, then the simplicial volume of $M$ is at most the volume of $M$ times a constant depending on $n$ and $V_1$.

Simplicial volume and 0-strata of separating filtrations

Abstract

We use Papasoglu's method of area-minimizing separating sets to give an alternative proof, and explicit constants, for the following theorem of Guth and Braun--Sauer: If is a closed, oriented, -dimensional manifold, with a Riemannian metric such that every ball of radius in the universal cover of has volume at most , then the simplicial volume of is at most the volume of times a constant depending on and .
Paper Structure (9 sections, 17 theorems, 48 equations, 1 figure)

This paper contains 9 sections, 17 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on simplicial volume
  3. Simplex straightening
  4. Triangulating a separating filtration
  5. Large-systole case
  6. Main proof
  7. The idea of Cantor bundles
  8. Adapting the proof to Cantor bundles
  9. Integral foliated simplicial volume

Key Result

Theorem 1

Let $M$ be a closed, oriented, connected $n$-dimensional Riemannian manifold, and let $\Gamma = \pi_1(M)$. Suppose that for all points $\widetilde{p}$ in the universal cover $\widetilde{M}$ of $M$, we have $\mathop{\mathrm{Vol}}\nolimits B(\widetilde{p}, 1) \leq V_1$. Then where $\Vert M \Vert_{\Delta}$ denotes the Gromov simplicial volume of $M$. Furthermore, if $V_1 < \frac{1}{n!}$, then the im

Figures (1)

  • Figure 1: In an $R$-separating filtration, $Z_0$ (shown as gray dots) separates $Z_1$ (shown as thick lines) into pieces of size at most $R$, and $Z_1$ separates $Z_2$ (shown as three planes) into pieces of size at most $R$. The smooth nesting condition implies that $Z_0$ avoids the points where multiple edges of $Z_1$ come together, and $Z_1$ does not run along the line where multiple planes of $Z_2$ come together.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2: gromov09
  • proof : Proof sketch
  • Corollary 3
  • proof : Proof sketch
  • Lemma 4
  • proof
  • Theorem 5
  • Lemma 6
  • proof
  • ...and 25 more