Distal actions on coset spaces in totally disconnected locally compact groups

Reid, Colin D.

doi:10.1142/s1793525319500523

Cited by 14 publications

(22 citation statements)

References 28 publications

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“…Since K generates G, we deduce that the compact open subgroup B φ t is normal in G, for each t ∈ ]0, r]. ✷ Related problems were also studied in [47].…”

Section: 6mentioning

confidence: 89%

Endomorphisms of Lie Groups over Local Fields

Glöckner

2018

MATRIX Book Series

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“…Since K generates G, we deduce that the compact open subgroup B φ t is normal in G, for each t ∈ ]0, r]. ✷ Related problems were also studied in [47].…”

Section: 6mentioning

confidence: 89%

Endomorphisms of Lie Groups over Local Fields

Glöckner

2018

MATRIX Book Series

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“…Combining Theorems 1.1 and 1.8, we obtain a negative answer to a question asked by Colin Reid [18,Question 2]; see Section 4.5. been a source of inspiration for the proof of Theorem 4. 16.…”

Section: Introductionmentioning

confidence: 85%

“…The relative Tits core of g in G, denoted by G † g , is defined by G † g := con(g) ∪ con(g −1 ) . Question 4.20 (Reid,[18,Question 2]). Let G be a t.d.l.c.s.c.…”

Section: Simple Subquotientsmentioning

confidence: 99%

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

Caprace

Wesolek

2018

Geom. Topol.

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We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite index subgroup which surjects onto Z. The second ensures that irreducible cocompact lattices in a product of non-discrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a general result, of independent interest, on irreducible lattices in product groups. The third implies that every non-discrete Burger-Mozes universal group of automorphisms of a tree with an arbitrary prescribed local action admits a compactly generated closed subgroup with a non-discrete simple quotient. As applications, we answer a question of D. Wise by proving the non-residual finiteness of a certain lattice in a product of two regular trees, and we obtain a negative answer to a question of C. Reid, concerning the structure theory of locally compact groups. arXiv:1708.04590v1 [math.GR]

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“…The result [7, Lemma 4.9] ensures that s • π =ŝ, hence s • φ =ŝ • β U , proving (1). Appealing again to [7,Lemma 4.9], if U is tidy for g inĜ U , then π(U ) is tidy for π(g) in H, and conversely if V is tidy for π(g) in H, then π −1 (V ) is tidy for g inĜ U . Therefore, if β U (X) has a common tidy subgroup, then so does φ(X).…”

Section: Invariant Properties Of Completionsmentioning

confidence: 96%

Homomorphisms into totally disconnected, locally compact groups with dense image

Reid

Wesolek

2019

Forum Mathematicum

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Let φ : G → H be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of φ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair (G, φ −1 (L)), where L is a compact open subgroup of H. These results generalize the well-known properties of profinite completions to the locally compact setting.All t.d.l.c. completions arise as completions with respect to a certain family of uniformities.Definition 1.2. Let G be a group and let S be a set of open subgroups of G. We say that S is a G-stable local filter if the following conditions hold:(a) S is non-empty; (b) Any two elements of S are commensurate; (c) Given a finite subset {V 1 , . . . , V n } of S, then n i=1 V i ∈ S, and given V ≤ W ≤ G such that |W : V | is finite, then V ∈ S implies W ∈ S; (d) Given V ∈ S and g ∈ G, then gV g −1 ∈ S.Each G-stable local filter S is a basis at 1 for a (not necessarily Hausdorff) group topology on G, and thus, there is an associated right uniformity Φ r (S) on G. The completion with respect to this uniformity, denoted byĜ S , turns out to be a t.d.l.c. group (Theorem 3.9); we denote by β G,S : G →Ĝ S the canonical inclusion homomorphism. All t.d.l.c. completions moreover arise in this way. Theorem 1.3 (see Theorem 4.3). If G is a topological group and φ : G → H is a t.d.l.c. completion map, then there is a G-stable local filter S and a unique topological group isomorphism ψ :We next consider completion maps φ : G → H where a specified subgroup U of G is the preimage of a compact open subgroup of H. In this case, there are two canonical completions of G such that U is the preimage of a compact open subgroup of the completion: the Belyaev completion, denoted byĜ U , and the Schlichting completion, denoted by G/ /U . These completions are the 'largest' and 'smallest' completions in the following sense.

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Distal actions on coset spaces in totally disconnected locally compact groups

Cited by 14 publications

References 28 publications

Endomorphisms of Lie Groups over Local Fields

Endomorphisms of Lie Groups over Local Fields

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

Homomorphisms into totally disconnected, locally compact groups with dense image

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