Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Le Conte de la Mesure sur les Complexes Cubiques CAT(0)

Talia Fernós

Abstract

We revisit the topic of probability measures on CAT(0) cube complexes and prove that an amenable group acting on a CAT(0) cube complex, regardless of dimension, necessarily preserves an interval in the Roller compactification. In the finite dimensional case, we prove that there must be an orbit of cardinality $2^N$, where $N$ is bounded by the dimension. This is a slight extension of the author's previous Tits' Alternative.

Le Conte de la Mesure sur les Complexes Cubiques CAT(0)

Abstract

We revisit the topic of probability measures on CAT(0) cube complexes and prove that an amenable group acting on a CAT(0) cube complex, regardless of dimension, necessarily preserves an interval in the Roller compactification. In the finite dimensional case, we prove that there must be an orbit of cardinality , where is bounded by the dimension. This is a slight extension of the author's previous Tits' Alternative.
Paper Structure (7 sections, 10 theorems, 9 equations, 11 figures)

This paper contains 7 sections, 10 theorems, 9 equations, 11 figures.

Table of Contents

  1. Functoriality: Walled Spaces
  2. The Roller Compactification
  3. Comparing halfspaces
  4. Intervals in $\overline{X}$
  5. The Helly Property in $\overline{X}$
  6. Probability Measures
  7. Medians

Key Result

Theorem 1

Let $X$ be a CAT(0) cube complex of finite dimension $D$, and suppose that $\Gamma\to \text{Aut}\, X$ preserves an interval in the Roller compactification $\overline{X}$. Then $\Gamma$ must have an orbit in $\overline{X}$ of cardinality $2^N$, for some $0\leqslant N\leqslant D$. If $X$ is not finite

Figures (11)

  • Figure 1: Making a cube complex CAT(0)
  • Figure 2: Functorial Construction
  • Figure 3: Two CAT(0) Cube Complexes with 10 halfspaces
  • Figure 4: An example where a section to the projection is impossible
  • Figure 5: Halfspaces containing $\infty_n$.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Theorem
  • Corollary : Tits' Alternative
  • Corollary
  • Lemma
  • Theorem
  • Theorem
  • Corollary
  • Lemma
  • Proposition
  • Lemma