Borel chromatic numbers of free groups
Abstract:
Recall that a coloring of a graph is a labeling of its vertices such that
no pair of vertices joined by an edge have the same label. The chromatic
number of a graph is the smallest number of colors for which there is a
coloring.
If G is a finitely generated group with generating set S, then for any free
action of G on a standard Borel space X, we can place a copy of the
S-Cayley graph of G onto every orbit. This results in a graph whose vertex
set is X and whose edge set is Borel measurable. We can then consider Borel
measurable colorings of this graph and the corresponding Borel chromatic
number. In this talk I will discuss Borel chromatic numbers for actions of
free groups with their standard generating set. I will focus on a
surprising theorem of Andrew Marks.