Indicability, residual finiteness, and simple subquotients of groups acting on trees

Caprace, Pierre–Emmanuel; Wesolek, Phillip

doi:10.2140/gt.2018.22.4163

Cited by 8 publications

(20 citation statements)

References 38 publications

(48 reference statements)

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“…The criterion developed by Burger-Mozes in order to prove their version of Proposition 4.16 ensures that, under suitable conditions on the local action, an irreducible lattice in Aut(T 1 )×Aut(T 2 ) whose projections to at least one of the factors is not injective, cannot be residually finite, see [BM00b, Proposition 2.1]. That criterion was subsequently generalized to lattices in products of CAT(0) spaces [CM12, Proposition 2.4], and then to irreducible lattices in products of locally compact groups in [CKRW, Corollary 33] and [CW,§5]. We record the following geometric version, where there is no condition on the local action.…”

Section: 4mentioning

confidence: 99%

Finite and Infinite Quotients of Discrete and Indiscrete Groups

Caprace¹

2019

Groups St Andrews 2017 in Birmingham

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These notes are devoted to lattices in products of trees and related topics. They provide an introduction to the construction, by M. Burger and S. Mozes, of examples of such lattices that are simple as abstract groups. Two features of that construction are emphasized: the relevance of non-discrete locally compact groups, and the two-step strategy in the proof of simplicity, addressing separately, and with completely different methods, the existence of finite and infinite quotients. A brief history of the quest for finitely generated and finitely presented infinite simple groups is also sketched. A comparison with Margulis' proof of Kneser's simplicity conjecture is discussed, and the relevance of the Classification of the Finite Simple Groups is pointed out. A final chapter is devoted to finite and infinite quotients of hyperbolic groups and their relation to the asymptotic properties of the finite simple groups. Numerous open problems are discussed along the way. arXiv:1709.05949v3 [math.GR]

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Section: 4mentioning

confidence: 99%

Finite and Infinite Quotients of Discrete and Indiscrete Groups

Caprace¹

2019

Groups St Andrews 2017 in Birmingham

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“…Since the subgroup G Σ Γ is dense in G by our assumption on the projection of Γ on G Π , it follows that Γ ∩ G Π is a discrete normal subgroup of G. Therefore it lies in the quasi-center QZ(G Π ). Notice that the quasi-center of a product group is the product of their quasi-centers (see [CW,Lemma 5.5]). So QZ(G Π ) is trivial, and so is Γ ∩ G Π .…”

Section: Relations Between the Irreducibility Conditionsmentioning

confidence: 99%

Bounding the covolume of lattices in products

Caprace¹,

Boudec²

2019

Compositio Math.

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We study lattices in a product G = G 1 ×· · ·×G n of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that G i is non-compact and every closed normal subgroup of G i is discrete or cocompact (e.g. G i is topologically simple).We show that the set of discrete subgroups of G containing a fixed cocompact lattice Γ with dense projections is finite. The same result holds if Γ is nonuniform, provided G has Kazhdan's property (T). We show that for any compact subset K ⊂ G, the collection of discrete subgroups Γ ≤ G with G = ΓK and dense projections is uniformly discrete, hence of covolume bounded away from 0. When the ambient group G is compactly presented, we show in addition that the collection of those lattices falls into finitely many Aut(G)-orbits.As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor.We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-noncompact tdlc group G is a Chabauty limit of discrete subgroups, then some compact open subgroup of G is an infinitely generated pro-p group for some prime p. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.

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“…For n ∈ N * ≥3 and Γ ≤ Sym(n), K (Γ) acts primitively on each of Br(D n ), Ends(D n ) and Reg(D n ). 5.F. Fixed points and sets.…”

Section: E Primitivitymentioning

confidence: 99%

Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms

2019

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Given a transitive permutation group, a fundamental object for studying its higher transitivity properties is the permutation action of its isotropy subgroup. We reverse this relationship and introduce a universal construction of infinite permutation groups that takes as input a given system of imprimitivity for its isotropy subgroup.This produces vast families of kaleidoscopic groups. We investigate their algebraic properties, such as simplicity and oligomorphy; their homological properties, such as acyclicity or contrariwise large Schur multipliers; their topological properties, such as unique polishability.Our construction is carried out within the framework of homeomorphism groups of topological dendrites.Date: January 2018. B.D. is supported in part by French projects ANR-14-CE25-0004 GAMME and ANR-16-CE40-0022-01 AGIRA. 1 Under the product topology, [n] D n is a Polish space, since D n is countable. The set of colorings C ⊆ [n] D n is easily verified to be a G δ set, so that it is a Polish space under the subspace topology; see for instance [16, Thm. 3.11]. When [n] is finite, C is in fact closed. Definition 3.2. A coloring of D n is kaleidoscopic if for all distinct x, y ∈ Br(D n ) and all distinct i, j ∈ [n], there is z ∈ (x, y) ∩ Br(D n ) such that c z (x) = i and c z (y) = j.In the following picture, open alcoves depict the two components U z (x) and U z (y) respectively colored with i and j.

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Indicability, residual finiteness, and simple subquotients of groups acting on trees

Cited by 8 publications

References 38 publications

Finite and Infinite Quotients of Discrete and Indiscrete Groups

Finite and Infinite Quotients of Discrete and Indiscrete Groups

Bounding the covolume of lattices in products

Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms

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