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## D. Gaboriau

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Initially: preparatory and personal notes for various Masterclasses
on Ergodic Theory, Measure Equivalence, Cost and L^2 Betti numbers
at various places.

Regular additions (2016, 2018, 2019, 2020,2022).

Trying to keep track of new developments in cost theory.

Document in rough draft form.

Comments welcome.

Download: lecture
notes in pdf

Download: Vademecum
on ℓ^{2} Betti numbers

## Abstract

The goal of this series of lectures is to present an
overview of the theory of orbit equivalence, with a particular
focus on the probability measure preserving actions of the free
groups. I will start by giving the basis of the theory of orbit
equivalence and explain the theory of cost. In particular, prove
such statements as the induction formula and the computation of
the cost of free actions of some countable groups, including free
groups. This will be related to the fundamental group of
equivalence relations. I intend to present Abert-Nikolov theorem
relating the cost of profinite actions to the rank gradient of the
associated chain of subgroups. I will consider a recent result of
F. Le Maître establishing a perfect connection between the cost
of a probability measure preserving action with the number of
topological generators of the associated full group. I shall also
discuss the number of non orbit equivalent actions of countable
groups.

* Earlier versions Copenhagen Lectures