Logic Seminar

Introduced by Levitt in 1995 the notion of cost turned out to be a
useful invariant for countable Borel equivalence relations in the presence of
a Borel probability measure which is invariant for the given equivalence
relation $E$. One prominent result is that when the cost of $E$ is greater than
1, then there is a Borel subequivalence relation induced by a free Borel
action of free group in two generators.

The goal of this talk is to present a generalization of this concept for an
arbitrary Borel cocycle. Among other results it will be shown that, given a
Borel probability measure which is invariant with respect to a given cocycle,
a countable Borel equivalence relation $E$ is hyperfinite on a conull
$E$-invariant Borel set iff the cost is attained and equals 1.