Invariant random subgroups over non-Archimedean local fields

Gelander, Tsachik; Levit, Arie

doi:10.1007/s00208-018-1767-8

Cited by 23 publications

(43 citation statements)

References 42 publications

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“…Local Rigidity Combining Theorem 7.2 and Proposition 7.9 of [GL17] we obtain the following extension of Theorem 1.9: Theorem 6. (Chabauty local rigidity [GL17]) Let G be a semisimple analytic group and Γ an irreducible lattice in G. If Γ is locally rigid then it is also Chabauty locally rigid.…”

Section: Self Chabauty Isolationmentioning

confidence: 82%

“…Hence, there is no problem in allowing X = H 3 in Corollary 14. The analog of Corollary 14 for p-adic Bruhat Tits buildings is proved in [GL17].…”

Section: Applications To L 2 -Invariantsmentioning

confidence: 89%

“…, U n ⊂ G so that the only closed subgroup intersecting every U i non-trivially is G itself. The following result is proved in [GL17,§6].…”

Section: Self Chabauty Isolationmentioning

confidence: 98%

“…Farber Property The proof presented above for Theorem 1.12 follows the lines developed at [GL17] and is simpler and applies to a more general setup than the original proof from [7s12]. In particular, the following general version of [7s12, Theorem 4.4] is proved in [GL17]:…”

Section: Self Chabauty Isolationmentioning

confidence: 99%

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A View on Invariant Random Subgroups and Lattices

Gelander¹

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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For more than half a century lattices in Lie groups played an important role in geometry, number theory and group theory. Recently the notion of Invariant Random Subgroups (IRS) emerged as a natural generalization of lattices. It is thus intriguing to extend results from the theory of lattices to the context of IRS, and to study lattices by analyzing the compact space of all IRS of a given group. This article focuses on the interplay between lattices and IRS, mainly in the classical case of semisimple analytic groups over local fields.

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Section: Self Chabauty Isolationmentioning

confidence: 82%

“…Hence, there is no problem in allowing X = H 3 in Corollary 14. The analog of Corollary 14 for p-adic Bruhat Tits buildings is proved in [GL17].…”

Section: Applications To L 2 -Invariantsmentioning

confidence: 89%

“…, U n ⊂ G so that the only closed subgroup intersecting every U i non-trivially is G itself. The following result is proved in [GL17,§6].…”

Section: Self Chabauty Isolationmentioning

confidence: 98%

Section: Self Chabauty Isolationmentioning

confidence: 99%

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A View on Invariant Random Subgroups and Lattices

Gelander¹

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…The study of invariant random subgroups on various classes of groups has been an active area of research in the last several years, see, for example, [BGK17] and the references contained therein, as well as [Bo14], [TT-D14], [LM15], [Ge15], [BGN15], [O15], [LM15], [GL16], [HT16], [EG16], [BDLW16], [G17], [BBT17], [BT17], [DM17], [HY17], [Ge18], [BT18], [TT-D18], [BLT18], [GeL18]. One usually concentrates on the study of ergodic (with respect to the conjugacy action) invariant random subgroups, which are the extreme points of the Choquet simplex IRS(Γ).…”

Section: Introductionmentioning

confidence: 99%

Co-induction and invariant random subgroups

Kechris

Quorning

2019

Groups Geom. Dyn.

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In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group.We use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of F 2 which are all invariant and weakly mixing with respect to the action of Aut(F 2 ).Moreover, for amenable groups Γ ≤ ∆, we obtain that the standard co-induction operation from the space of weak equivalence classes of Γ to the space of weak equivalence classes of ∆ is continuous if and only if [∆ : Γ] < ∞ or core ∆ (Γ) is trivial. For general groups we obtain that the co-induction operation is not continuous when [∆ : Γ] = ∞. This answers a question raised by Burton and Kechris in [BK18]. Independently such an answer was also obtained, using a different method, by Bernshteyn in [B18].

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Critical exponents of invariant random subgroups in negative curvature

Gekhtman

Levit

2019

Geom. Funct. Anal.

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Let X be a proper geodesic Gromov hyperbolic metric space and let G be a cocompact group of isometries of X admitting a uniform lattice. Let d be the Hausdorff dimension of the Gromov boundary ∂X. We define the critical exponent δ(µ) of any discrete invariant random subgroup µ of the locally compact group G and show that δ(µ) > d 2 in general and that δ(µ) = d if µ is of divergence type. Whenever G is a rank-one simple Lie group with Kazhdan's property (T ) it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo.

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Invariant random subgroups over non-Archimedean local fields

Cited by 23 publications

References 42 publications

A View on Invariant Random Subgroups and Lattices

A View on Invariant Random Subgroups and Lattices

Co-induction and invariant random subgroups

Critical exponents of invariant random subgroups in negative curvature

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