Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integral foliated simplicial volume and ergodic decomposition

Clara Loeh, Giovanni Sartori

Abstract

We establish an integration formula for integral foliated simplicial volume along ergodic decompositions. This is analogous to the ergodic decomposition formula for the cost of groups.

Integral foliated simplicial volume and ergodic decomposition

Abstract

We establish an integration formula for integral foliated simplicial volume along ergodic decompositions. This is analogous to the ergodic decomposition formula for the cost of groups.
Paper Structure (11 sections, 14 theorems, 56 equations)

This paper contains 11 sections, 14 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Organisation of this paper
  3. Integral foliated simplicial volume
  4. Basic definitions
  5. A strict version
  6. Ergodic decomposition
  7. Proof of Theorem \ref{['thm:ifsvergdec']}
  8. Reduction: countably many simplices
  9. Reduction: countably many functions
  10. Reduction: countably many parametrised chains
  11. Proof of the ergodic decomposition formula

Key Result

Theorem 1.1

Let $M$ be an oriented closed connected manifold with fundamental group $\Gamma$, let $(\alpha,\mu)\colon\Gamma \curvearrowright X$ be a standard probability action, and let $\beta \colon X \rightarrow \mathop{\mathrm{Erg}}\nolimits(\alpha)$ be an ergodic decomposition of $(\alpha,\mu)$. Then

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Proposition 1.3: loehpagliantini
  • Definition 1.4: fixed price loeh_cost*Definition 1.3
  • Definition 2.1: standard (probability) actions and bounded functions
  • Definition 2.2: parametrised fundamental cycles
  • Definition 2.3: integral foliated simplicial volume
  • Proposition 2.4: comparison with integral and real simplicial volume loehpagliantini
  • Definition 2.5: bounded functions
  • ...and 29 more