Stabilizer rigidity in irreducible group actions

Hartman, Yair; Tamuz, Omer

doi:10.1007/s11856-016-1424-4

Cited by 13 publications

(12 citation statements)

References 27 publications

(52 reference statements)

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“…The following lemma yields the existence of a minimal closed subgroup in which an IRS lies. The existence of such minimal subgroup was already proved in [HT13]. This subgroup is called the normal closure of the IRS.…”

Section: Spanning Irssmentioning

confidence: 98%

Amenable invariant random subgroups

et al. 2016

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Section: Spanning Irssmentioning

confidence: 98%

Amenable invariant random subgroups

et al. 2016

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“…In fact, the possibility of the current generalization is already suggested there (see Section 5.3 of [23]). Therefore our contribution to Theorem 1.2 is little more than its restatement in the above form, and the application of Theorem 1.1.Let us note that the Stuck-Zimmer theorem has been recently extended in the above mentioned work [7] as well as in [13].Remark. G k has property (T) whenever rank k (H) = 1 for every almost k-simple subgroup H of G, and this is in fact a necessary condition unless k is R or C.1.3.…”

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confidence: 91%

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

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confidence: 95%

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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“…Although the stationary spaces that we construct in this paper are all Poisson bundles, we do not elaborate regarding this correspondence, as this will not be important for understanding our results. Rather, we refer the interested reader to [HT16, Bow14, HT15, Kai05], and mention here the important features of Poisson bundles that are relevant to our context. To describe these features, we now turn to describe Schreier graphs.…”

Section: Preliminariesmentioning

confidence: 99%

Furstenberg entropy of intersectional invariant random subgroups

Hartman¹,

Yadin

2018

Compositio Math.

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We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a-priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen.The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups.For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs.

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Stabilizer rigidity in irreducible group actions

Cited by 13 publications

References 27 publications

Amenable invariant random subgroups

Amenable invariant random subgroups

The Nevo–Zimmer intermediate factor theorem over local fields

Furstenberg entropy of intersectional invariant random subgroups

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