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Locally integrable cross sections and their intersection covolume

Nachi Avraham-Re'em, Michael Björklund, Rickard Cullman

Abstract

We study systematically cross sections of probability preserving actions of unimodular groups and their associated transverse measures, and introduce the invariant \emph{intersection covolume} to quantify their periodicity. Our main theorem, derived from a higher order version of Kac's lemma, shows that the intersection covolume is bounded below by the intensity, with equality precisely when the action is induced by a lattice (in the sense of Mackey). We further prove that the natural cross sections of cut--and--project actions have finite intersection covolume.

Locally integrable cross sections and their intersection covolume

Abstract

We study systematically cross sections of probability preserving actions of unimodular groups and their associated transverse measures, and introduce the invariant \emph{intersection covolume} to quantify their periodicity. Our main theorem, derived from a higher order version of Kac's lemma, shows that the intersection covolume is bounded below by the intensity, with equality precisely when the action is induced by a lattice (in the sense of Mackey). We further prove that the natural cross sections of cut--and--project actions have finite intersection covolume.

Paper Structure

This paper contains 24 sections, 32 theorems, 216 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cross sections, intensity and transverse measures
  3. Periodicity and intersection covolume
  4. Cut--and--project actions and their cross sections
  5. Groups without lattices
  6. General preliminaries
  7. Mecke equation and relative invariance
  8. Background
  9. Basic definitions
  10. The transverse correspondence
  11. Proof of Campbell theorem
  12. Injective covers
  13. Final proof of the transverse correspondence
  14. Transverse $G$-spaces and $G$-factors
  15. Null sets in transverse $G$-spaces
  16. ...and 9 more sections

Key Result

Theorem A

For every probability preserving $G$-space $\left(X,\mu\right)$ with a locally integrable cross section $Y$, with equality if and only if there is a transverse $G$-factor from $\left(X,\mu,Y\right)$ onto a homogeneous space $\left(\Gamma\backslash G,m_{\Gamma\backslash G},\{\Gamma\}\right)$ for a lattice $\Gamma<G$.

Figures (2)

  • Figure 1: The dashed arrows are formed by the transverse correspondence \ref{['thm:corresp']}, with the upper space being transverse $G$-space and the lower being the associated transverse measure.
  • Figure 2: The suspension flow on $Z=\left[0,1\right]$ under the function $r=2$ on $Z_{\epsilon}=\left[0.9,1\right]$ (for $\epsilon=0.1$) and $r=1$ elsewhere, and the cross section $Y=\left[0,1\right]\times\{0\}$.

Theorems & Definitions (71)

  • Theorem A
  • Corollary 1.1
  • Theorem B
  • Theorem
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Theorem 3.4
  • Theorem 3.5: Lusin--Novikov
  • Lemma 3.7
  • ...and 61 more