SOME COMPUTATIONS OF 1-COHOMOLOGY GROUPS AND CONSTRUCTION OF NON-ORBIT-EQUIVALENT ACTIONS

By Sorin Popa. Journal of the Institute of Mathematics of Jussieu, 2004

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For each group $G$ having an infinite normal subgroup with the relative property (T) (e.g. $G=H\times K$, with $H$ infinite with property (T) and $K$ arbitrary) and each countable abelian group $\varLambda$ we construct free ergodic measure-preserving actions $\sigma_\varLambda$ of $G$ on the probability space such that the first cohomology group of $\sigma_\varLambda$, $\ssm{H}^1(\sigma_\varLambda,G)$, is equal to $\text{Char}(G)\times\varLambda$. We deduce that $G$ has uncountably many non-stably orbit-equivalent actions. We also calculate 1-cohomology groups and show existence of ‘many’ non-stably orbit-equivalent actions for free products of groups as above.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c0143cb8e4c9490d7a7306f463d57c54d19f7779

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Citations in the corpus

On Popa's Cocycle Superrigidity Theorem (2006, background): “This theorem continues a series of ground breaking results in the area of von Neumann Algebras recently obtained by Sorin Popa [27], [25], [23], [24], [26], and by Popa and collaborators: Popa-Sasyk [28], Ioana-Peterson-Popa [16], Popa-Vaes [29].”
Orbit Equivalence and Measured Group Theory (2010, background): “And along the years, various rigidity results entailed some speciﬁc families of groups to admit continously 7 many non-OE actions (see for instance [BG81, Zim84, GG88b, MS06, Pop06b, Pop07a]).”
Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups (2006, background): “The inner automorphism group InnR, also called full group ofR, consists of ∆ ∈ Aut(X,µ) such that (x,∆(x)) ∈ R for almost all x ∈ X .”
Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups (2005, background): “6.8. Let { i}i∈I be a countable (at most) family of groups such that each i has a wq-normal rigid subgroup Hi ⊂ i. Denote = ∗i i the free product of the groups i. As pointed out in ([ P4 ]), if |I| ≥ 2 then does not have any wq-normal rigid subgroup.”
Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I (2003, background): “Note that Λ is wqnormal in Γ if and only if there exists an ordinal i and a well ordered family of intermediate subgroups Λ = H0 ⊂ H1 ⊂ ... ⊂ Hj ⊂ ... ⊂ Hi = Γ such that for each j Po6 ]).”
Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa) (2006, methodology): “This result allows to compute as well the 1cohomology for quotients of Bernoulli actions, yielding the following result of [51].”
GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS (2017, background): “Property (T) has very strong consequences for the dynamics of group actions in different settings; see for instance [6, 48, 119, 125, 138].”
On the superrigidity of malleable actions with spectral gap (2006, background): “By 2.7 in [ P2 ], Corollary 1.2 provides a large class of groups with uncountably many OE inequivalent actions, adding to the numerous examples already found in [Z2], [H], [GP], [P2], [I1], [IPeP]:”
Orbit inequivalent actions for groups containing a copy of $\mathbb{F}_{2}$ (2006, background): “Proof. Since α is weakly mixing (see for example [ P2 ]), Lemma 2.2.(ii) implies that ˜ α is weakly mixing.”
On cocycle superrigidity for Gaussian actions (2009, background): “The notions of wq-normality and the class Ufin are due to Popa (cf. [35, 38]).”
Group cocycles and the ring of affiliated operators (2007, methodology): “required normality property. Clearly, normal implies s-normal implies q-normal; and similarly for the weak notions. The notion of (weakly) q-normal subgroups were introduced by Popa (Deﬁnition 2.3 in [38]) in order to “untwist” cocycles from the subgroup to the whole group under certain weak mixingn”