Balanced Measures on Compact Median Algebras

TL;DR

This work classifies balanced measures for the self-median dynamics on sclocc median algebras, showing that balanced measures are always cubical with support a cube and that the uniform cube measure is the unique fully supported balanced measure. It deduces that, for amenable group actions, there exists an invariant cube, linking dynamics on median algebras to cube structures and Roller compactifications. The results hinge on analyzing half-spaces, walls, and gate-projections to constrain measure supports, culminating in a clean cubical dichotomy that generalizes to CAT(0) cube complexes via Roller compactifications. Overall, the paper reveals that balanced invariant dynamics on these spaces reduce to cube-structured components, with precise uniqueness and fixed-point implications for amenable actions.

Abstract

We initiate a systematic investigation of group actions on compact medain algebras via the corresponding dynamics on their spaces of measures. We show that a probability measure which is invariant under a natural push forward operation must be a uniform measure on a cube and use this to show that every amenable group action on a locally convex compact median algebra fixed a sub-cube.

Theorems & Definitions (31)

• Theorem A
• Proposition 1.1
• Proposition 1.2
• Corollary B
• Lemma 2.1: statements (Int 3) and (Int 6) at the beginning of section 2 in rol98
• Lemma 2.2: rol98
• Lemma 2.3: fio20
• Lemma 2.4
• proof
• Lemma 2.5