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...is virtually isomorphic to a lattice Λ1 in Ad G, and (G/Γ × X,G) is virtually isomorphic to (Ad G/Λ1 × Y1,Ad G) (see [Zi3]). Another fact, related to our discussion, is the result by Stuck and Zimmer =-=[SZ]-=-, which states that any f.m.p. nonatomic ergodic action of a higher rank lattice is essentially free, up to a division by the finite center.ORBIT EQUIVALENCE RIGIDITY 1085 In the sequel we use a noti...

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...or all but the third arrow it is easy to find examples showing that the reverse implications do not hold. We shall discuss these classes more in Section 4. Note that a deep result of Stuck and Zimmer =-=[30]-=- tells us that every faithful ergodic measure preserving action of a higher rank semisimple real lattice on a probability space is essentially free. In particular, every faithful boundary representati...

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... ergodically on (X, µ). Remarks. (a) Every mixing ergodic action (X, µ, Γ) of an irreducible lattice Γ in a semi-simple Lie group G is irreducible and aperiodic. (b) By the result of Stuck and Zimmer =-=[14]-=- any ergodic non-atomic m.p. action of an irreducible lattice Γ in a semi-simple Lie group G with property (T) is free (mod 0). Recall that a higher rank semi-simple G has property (T) iff it does not...

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...actions, introduced by Zimmer in an earlier work [118], became an indispensable tool in the field. Zimmer continued to study orbit structures of actions of large groups in [25,34,120–124,126,127] and =-=[115]-=-. The monograph [126] had a particularly big impact on both ergodic theorists and people studying big groups, as well as researchers in other areas, such as Operator Algebras and Descriptive Set Theor...

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...eneralized Bernoulli action, and (X, µ)−→(Y, ν) a measurable quotient where each g ∈ Γ acts with finite entropy. Then (Y, ν) is a point. This follows from the following facts: (1) By Stuck and Zimmer =-=[31]-=- an ergodic action of a higher rank lattice is essentially free, unless the action is transitive one on a finite set. Since Γ � (X, µ) is weakly mixing, it follows that Γ � (Y, ν) is essentially free....

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... µΓ where Γ is a lattice in G. • The set of ergodic IRS’s is compact and its only accumulation point is µId. The first part of Theorem 1.14 is a consequence of the Nevo–Stück–Zimmer rigidity theorem =-=[110, 88]-=-. The picture is much wilder in rank one. For example, starting with a lattice Γ ≤ G and an infinite index normal subgroup ∆ ⊳ Γ, one can induce the measure on Γ\G to an ergodic IRS supported on the c...

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... studies of the action of G on its space of subgroups [AGV12, Vo12, AB+11, Ve11, Sa11, Gr11, Ve10, Bo10, BS06, DS02, GS99, SZ94]. Perhaps the deepest result in the subject is the Stuck-Zimmer Theorem =-=[SZ94]-=-: there are no nonobvious IRS’s of higher rank simple Lie groups. More precisely, every ergodic IRS is induced from a lattice subgroup. Another nice result in this area is a complete classification IR...

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...invariant under conjugation is called an invariant random subgroup or IRS. One of the goals of this research is to classify the distributions of IRS’s for interesting groups. For example Stuck-Zimmer =-=[SZ94]-=- proved that every ergodic IRS of a higher rank simple Lie group is induced from a lattice subgroup. A complete classification of IRS’s of the infinite symmetric group has been obtained by A. Vershik ...