Groups, Geometry, and Dynamics


Full-Text PDF (440 KB) | Metadata | Table of Contents | GGD summary
Volume 7, Issue 1, 2013, pp. 127–180
DOI: 10.4171/GGD/179

Published online: 2013-01-31

Measurable chromatic and independence numbers for ergodic graphs and group actions

Clinton T. Conley[1] and Alexander S. Kechris[2]

(1) Cornell University, Ithaca, USA
(2) California Institute of Technology, Pasadena, United States

We study in this paper combinatorial problems concerning graphs generated by measure preserving actions of countable groups on standard measure spaces. In particular we study chromatic and independence numbers, in both the measure-theoretic and the Borel context, and relate the behavior of these parameters to properties of the acting group such as amenability, Kazhdan’s property (T), and freeness. We also prove a Borel analog of the classical Brooks’ Theorem in finite combinatorics for actions of groups with finitely many ends.

Keywords: Ergodic actions, Borel combinatorics, chromatic numbers, amenable groups, free groups

Conley Clinton, Kechris Alexander: Measurable chromatic and independence numbers for ergodic graphs and group actions. Groups Geom. Dyn. 7 (2013), 127-180. doi: 10.4171/GGD/179