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TOTALLY NONFREE ACTIONS AND THE INFINITE SYMMETRIC GROUP
, 2011
"... To the memory of my beloved Lyotya We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjointinvariant and TNFmeasures on the lattice of subgro ..."
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To the memory of my beloved Lyotya We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjointinvariant and TNFmeasures on the lattice of subgroups of the infinite symmetric group SN. The problem is closely related to the theory of characters and factor representations of groups. 1
FURSTENBERG ENTROPY REALIZATIONS FOR VIRTUALLY FREE GROUPS AND LAMPLIGHTER GROUPS
"... Abstract. Let (G, µ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε> 0 such that the Furstenberg entropy of any (G, µ)stationary space is either zero or larger than ε. Virtually free groups, such as SL2(Z), do not have prop ..."
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Abstract. Let (G, µ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε> 0 such that the Furstenberg entropy of any (G, µ)stationary space is either zero or larger than ε. Virtually free groups, such as SL2(Z), do not have property (T). For these groups, we construct stationary actions with arbitrarily small, positive entropy. This construction involves building and lifting spaces of lamplighter groups. For some classical lamplighters, these spaces realize a dense set of entropies
STABILIZER RIGIDITY IN IRREDUCIBLE GROUP ACTIONS
"... Abstract. We consider irreducible actions of locally compact product groups, and of higher rank semisimple Lie groups. Using the intermediate factor theorems of BaderShalom and NevoZimmer, we show that the action stabilizers, and hence all irreducible invariant random subgroups, are coamenable i ..."
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Abstract. We consider irreducible actions of locally compact product groups, and of higher rank semisimple Lie groups. Using the intermediate factor theorems of BaderShalom and NevoZimmer, we show that the action stabilizers, and hence all irreducible invariant random subgroups, are coamenable in some normal subgroup. As a consequence, we derive rigidity results on irreducible actions that provide generalizations, and new proofs,
WEAK EQUIVALENCE AND NONCLASSIFIABILITY OF MEASURE PRESERVING ACTIONS
"... Abstract. AbértWeiss have shown that the Bernoulli shift sΓ of a countably infinite group Γ is weakly contained in any free measure preserving action a of Γ. Proving a conjecture of Ioana we establish a strong version of this result by showing that sΓ × a is weakly equivalent to a. Using random Ber ..."
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Abstract. AbértWeiss have shown that the Bernoulli shift sΓ of a countably infinite group Γ is weakly contained in any free measure preserving action a of Γ. Proving a conjecture of Ioana we establish a strong version of this result by showing that sΓ × a is weakly equivalent to a. Using random Bernoulli shifts introduced by AbértGlasnerVirag we generalized this to nonfree actions, replacing sΓ with a random Bernoulli shift associated to an invariant random subgroup, and replacing the product action with a relatively independent joining. The result for free actions is used along with the theory of Borel reducibility and Hjorth’s theory of turbulence to show that the equivalence relations of isomorphism, weak isomorphism, and unitary equivalence on the weak equivalence class of a free measure preserving action do not admit classification by countable structures. This in particular shows that there are no free weakly rigid actions, i.e., actions whose weak equivalence class and isomorphism class coincide, answering negatively a question of Abért and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. A countably infinite residually finite group Γ is said to have property EMD ∗ if the action pΓ of Γ on its profinite completion weakly contains all ergodic measure preserving actions of Γ, and Γ is said to have property MD if ι×pΓ weakly contains all measure preserving actions of Γ, where ι denotes the identity action on a standard nonatomic probability space. Kechris shows that EMD ∗ implies MD and asks if the two properties are actually equivalent. We provide a positive answer to this question by studying the relationship between convexity and weak containment in the space of measure preserving actions.
UNIMODULARITY OF INVARIANT RANDOM SUBGROUPS
"... Abstract. An invariant random subgroup H ≤ G is a random closed subgroup whose law is invariant to conjugation by all elements of G. When G is locally compact and second countable, we show that for every invariant random subgroup H ≤ G there almost surely exists an invariant measure on G/H. Equival ..."
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Abstract. An invariant random subgroup H ≤ G is a random closed subgroup whose law is invariant to conjugation by all elements of G. When G is locally compact and second countable, we show that for every invariant random subgroup H ≤ G there almost surely exists an invariant measure on G/H. Equivalently, the modular function of H is almost surely equal to the modular function of G, restricted to H. We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact invariant random subgroups. 1.
AND THE INFINITE SYMMETRIC GROUP
, 2011
"... To the memory of my beloved Lyotya We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjointinvariant andTNFmeasuresonthelatticeofsubgroupsoft ..."
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To the memory of my beloved Lyotya We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjointinvariant andTNFmeasuresonthelatticeofsubgroupsoftheinfinitesymmetricgroup SN. The problem is closely related to the theory of characters and factor representations of groups. 1