Course: Topics in Measurable Group Theory
Course Description: This course will introduce Gromov’s notion of Measure Equivalence (ME) of groups. This notion is the measurable group theoretic counterpart to geometric group theory’s Quasi Isometry (QI). We will cover properties which are preserved by ME such as amenability, the Haagerup property, and property (T), and other invariants which are sensitive to ME such as cost of groups and L^2-cohomology of groups and groupoids. We will cover groups which are ME to free groups (so-called treeable groups), ME-rigidity, Popa’s cocycle superrigidity theorems, and
Gaboriau and Lyons’s measurable group theoretic solution to von Neumann’s problem: every nonamenable group “measurably contains” a free group on two generators.
Background: Measurable group theory seeks to understand countably inﬁnite groups,
and more generally locally compact groups, through ergodic theoretic properties of their mea surable actions on standard probability spaces, and particularly through structural properties of the orbit equivalence relations generated by these actions. The motivating phenomenon is that algebraic properties of an acting group are often expressed through measurable properties of the associated equivalence relation. An extreme form of this phenomenon, going back to Zimmer’s groundbreaking work on ergodic actions of higher rank semisimple lie groups, is seen in orbit equivalence and cocycle superrigidity theorems, which state that in certain settings an equivalence relation completely remembers the group from which it was generated. At the other extreme are what might be called orbit equivalence anti-rigidity theorems, stating that certain groups and actions cannot be distinguished by looking at the equivalence relations they generate; this second extreme is exempliﬁed by the celebrated Ornstein-Weiss and Connes-Feldman-Weiss theorems which imply that any equivalence relation generated by an amenable group can also be generated, modulo a null set, by the group Z of integers.