Stabilizers of Ergodic Actions of Lattices and Commensurators

Creutz, Darren; Peterson, Janice

doi:10.48550/arxiv.1303.3949

Cited by 5 publications

(28 citation statements)

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“…The real Lie group case is discussed in detail in Subsection 7.3. such that p • f is the projection on X (for a general formulation of this argument, see Proposition 2.4.5 of [7]). Furthermore f and p are G-maps with respect to the measures µ × η, λ and η on G /P × X, Y and X respectively.…”

Section: The Stuck-zimmer Theoremmentioning

confidence: 99%

“…Let us note that the Stuck-Zimmer theorem has been recently extended in the above mentioned work [7] as well as in [13].…”

Section: Introductionmentioning

confidence: 96%

“…We mention that the Nevo-Zimmer intermediate factor theorem has been extended in several directions since it first appeared; see for example the works of Bader-Shalom [4], Creutz-Peterson [7] and the above mentioned [21].…”

Section: Introductionmentioning

confidence: 99%

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The Nevo–Zimmer intermediate factor theorem over local fields

Levit

2016

Geom Dedicata

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The Nevo-Zimmer theorem classifies the possible intermediate Gfactors Y in X × G /P → Y → X, where G is a higher rank semisimple Lie group, P a minimal parabolic and X an irreducible G-space with an invariant probability measure.An important corollary is the Stuck-Zimmer theorem, which states that a faithful irreducible action of a higher rank Kazhdan semisimple Lie group with an invariant probability measure is either transitive or free, up to a null set.We present a different proof of the first theorem, that allows us to extend these two well-known theorems to linear groups over arbitrary local fields.1 Subsection 2.2 contains definitions concerning algebraic k-groups, as well as some examples. 2 The reader is referred to Subsection 2.1 for a discussion of measurable group actions, including the definitions of a G-space and a G-map. 1 THE NEVO-ZIMMER INTERMEDIATE FACTOR THEOREM OVER LOCAL FIELDS 2of Furstenberg such as the interplay between stationary and P -invariant measures, and diverges from the original approach of Margulis in [17].We mention that a corrected proof along the lines of [25] was given by Nevo and Zimmer in the unpublished work [19].Our chief motivation is to present a proof of Theorem 1.1 that is as close as possible to the lines of [25] and [17], while extending the result to local fields. Indeed, the factor theorem [17] is already given in this (indeed, even greater) generality. We remark that our approach differs from that of [19].Indeed, the real case of Theorem 1.1 implies the real Lie group version at least for center-free semisimple Lie groups, and the same remark applies to Theorem 1.2. This reduction is based on the fact that every connected semisimple real Lie group with trivial center is isomorphic to G 0 R for some connected but possibly non simply-connected algebraic R-group. See Subsection 7.3 for further discussion.An important corollary of the intermediate factor theorem is the theorem of Stuck and Zimmer discussed in Subsection 1.2 below. Moreover the factor theorem is a consequence of the intermediate factor theorem, and the normal subgroup theorem for groups having property (T ) is a consequence of the Stuck-Zimmer theorem. So these four results are related, implying and generalizing each other.We mention that the Nevo-Zimmer intermediate factor theorem has been extended in several directions since it first appeared; see for example the works of Bader-Shalom [4], Creutz-Peterson [7] and the above mentioned [21].1.2. The Stuck-Zimmer theorem. The following is a formulation of this theorem in the setting of linear algebraic groups over local fields.Theorem 1.2. Assume that rank k (G) ≥ 2 and G k has property (T). Then every faithful, properly ergodic, irreducible and finite measure preserving action of G k is essentially free.More generally, if the action has central kernel (but is possibly not faithful) then µ-almost every point has central stabilizer.For example as G = SL n is simply-connected the above theorem holds for the groups SL n (R), SL n (Q p ) and SL n (F q ((t)))...

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Section: The Stuck-zimmer Theoremmentioning

confidence: 99%

“…Let us note that the Stuck-Zimmer theorem has been recently extended in the above mentioned work [7] as well as in [13].…”

Section: Introductionmentioning

confidence: 96%

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The Nevo–Zimmer intermediate factor theorem over local fields

Levit

2016

Geom Dedicata

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“…Creutz and Peterson [7] prove similar rigidity results for irreducible lattices and commensurators of lattices in semi-simple Lie groups, and also for product groups with the Howe-Moore property and property (T).…”

Section: Introductionmentioning

confidence: 56%

“…An invariant random subgroup (IRS) of G is a random variable that takes values in Sub G , the space of closed subgroups of G, and whose distribution is invariant to conjugation by any element of G [2]. IRSs arise naturally as stabilizers of probability measure preserving (pmp) actions, and in fact any IRS is the stabilizer of some pmp action (see [1,Theorem 2.4] and also [2,7]). They are also an interesting object of study as stochastic generalizations of normal subgroups, and of lattices.…”

Section: Introductionmentioning

confidence: 99%

Stabilizer rigidity in irreducible group actions

Hartman

Tamuz

2016

Isr. J. Math.

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Abstract. We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader-Shalom and NevoZimmer, we show that the action stabilizers, and all irreducible invariant random subgroups, are co-amenable in their normal closure. As a consequence, we derive rigidity results on irreducible actions that generalize and strengthen the results of Bader-Shalom and Stuck-Zimmer.

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Unimodularity of invariant random subgroups

Biringer

Tamuz

2016

Trans. Amer. Math. Soc.

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An invariant random subgroup H ≤ G H \leq G is a random closed subgroup whose law is invariant to conjugation by all elements of G G . When G G is locally compact and second countable, we show that for every invariant random subgroup H ≤ G H \leq G there almost surely exists an invariant measure on G / H G/H . Equivalently, the modular function of H H is almost surely equal to the modular function of G G , restricted to H 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.

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Stabilizers of Ergodic Actions of Lattices and Commensurators

Cited by 5 publications

References 34 publications

The Nevo–Zimmer intermediate factor theorem over local fields

The Nevo–Zimmer intermediate factor theorem over local fields

Stabilizer rigidity in irreducible group actions

Unimodularity of invariant random subgroups

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