Unique Cartan decomposition for II_<i>1</i> factorsarising from arbitrary actions of hyperbolic groups

Abstract

Abstract We prove that for any free ergodic probability measure preserving action <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Γ</m:mi> <m:mo>→</m:mo> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> $\Gamma \rightarrow (X,\mu )$ of a non-elementary hyperbolic group, or a lattice in a rank one simple Lie group, the associated group measure space II 1 factor <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mtext>L</m:mtext> </m:mrow> <m:mi>∞</m:mi> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>⋊</m:mo> <m:mi>Γ</m:mi> </m:mrow> </m:math> ${\mathord {\text{\rm L}}^\infty (X) \rtimes \Gamma }$ has <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mtext>L</m:mtext> </m:mrow> <m:mi>∞</m:mi> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${\mathord {\text{\rm L}}^\infty (X)}$ as its unique Cartan subalgebra, up to unitary conjugacy.