Clinton T. Conley

Ultraproducts of measure preserving actions and graph combinatorics

Ultraproducts of measure preserving actions of countable groups are used to study the graph combinatorics associated with such actions, including chromatic, independence and matching numbers. Applications are also given to the theory of random colorings of Cayley graphs and sofic actions and equivalence relations.

Measurable chromatic and independence numbers for ergodic graphs and group actions

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.

BROOKS’ THEOREM FOR MEASURABLE COLORINGS

We generalize Brookss theorem to show that if

An antibasis result for graphs of infinite Borel chromatic number

G

Følner tilings for actions of amenable groups

is a Borel graph on a standard Borel space

Measure reducibility of countable Borel equivalence relations

X

Measurable perfect matchings for acyclic locally countable borel graphs

of degree bounded by

Incomparable actions of free groups

d \geq 3

Stationary probability measures and topological realizations

which contains no

Definability of small puncture sets

(d+1)