# Relative Property (T) Actions and Trivial Outer
Automorphism
Groups

## D. Gaboriau

###

### **Journal of Functional Analysis, **260 (2011), no. 2,
414-427.

The original publication is available at Journal
of
Functional Analysis

doi:10.1016/j.jfa.2010.09.013

### Download: pdf (460K)

## Abstract

We show that every non-amenable free product of groups admits free
ergodic probability measure preserving actions which have relative
property (T)
in the sense of S. Popa (Definition 4.1 of [Pop06]). There are
uncountably
many such actions up to orbit equivalence and von Neumann
equivalence,
and
they may be chosen to be conjugate to any prescribed action when
restricted
to the free factors. We exhibit also, for every non-amenable free
product
of groups, free ergodic probability measure preserving actions
whose
associated
equivalence relation has trivial outer automorphisms group.
This
gives
in particular the first examples of such actions for the free
group on
2
generators.