Trees, contraction groups, and Moufang sets

Caprace, Pierre–Emmanuel; Medts, Tom De

doi:10.1215/00127094-2371640

Cited by 8 publications

(3 citation statements)

References 32 publications

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“…In simple Lie groups or in simple algebraic groups over local fields, the contraction group of every element is known to coincide with the unipotent radical of some parabolic subgroup, and is thus always closed. On the other hand, recent results indicate that many non-linear examples of groups in S have non-closed contraction groups (see [7] and [38]), while in some specific cases the closedness of contraction groups yields connections with finer algebraic structures related to linear algebraic groups (see [18]).…”

Section: Contraction Groups and Abstract Simplicitymentioning

confidence: 99%

Locally Normal Subgroups of Totally Disconnected Groups. Part Ii: Compactly Generated Simple Groups

Caprace

Reid

Willis

2017

Forum of Mathematics, Sigma

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We use the structure lattice, introduced in Part I, to undertake a systematic study of the class S consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are nondiscrete. Given G ∈ S , we show that compact open subgroups of G involve finitely many isomorphism types of composition factors, and do not have any soluble normal subgroup other than the trivial one. By results of Part I, this implies that the centralizer lattice and local decomposition lattice of G are Boolean algebras. We show that the G-action on the Stone space of those Boolean algebras is minimal, strongly proximal, and microsupported. Building upon those results, we obtain partial answers to the following key problems: Are all groups in S abstractly simple? Can a group in S be amenable? Can a group in S be such that the contraction groups of all of its elements are trivial? 2010 Mathematics Subject Classification: 22D05 (primary); 20E15, 20E32 (secondary)

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Section: Contraction Groups and Abstract Simplicitymentioning

confidence: 99%

Locally Normal Subgroups of Totally Disconnected Groups. Part Ii: Compactly Generated Simple Groups

Caprace

Reid

Willis

2017

Forum of Mathematics, Sigma

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“…On the other hand the full automorphism group of a d-regular tree is not linear, and has a simple subgroup of index 2 [Tit70]. We refer the reader to [CDM12] for more on linearity. There are also groups G in T which do not admit any free lattice.…”

Section: Introductionmentioning

confidence: 99%

Locally compact convergence groups and $$n$$ n -transitive actions

Carette

Dreesen

2014

Math. Z.

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All σ-compact, locally compact groups acting sharply n-transitively and continuously on compact spaces M have been classified, except for n = 2, 3 when M is infinite and disconnected. We show that no such actions exist for n = 2 and that these actions for n = 3 coincide with the action of a hyperbolic group on a space equivariantly homeomorphic to its hyperbolic boundary. We further give a characterization of non-compact groups acting 3-properly and transitively on infinite compact sets as non-elementary boundary transitive hyperbolic groups. The main tool is a generalization to locally compact groups of Bowditch's topological characterization of hyperbolic groups. Finally, in contrast to the case n = 3, we show that for n ≥ 4 if a locally compact group acts continuously, n-properly and n-cocompactly on a locally connected metrizable compactum M , then M has a local cut point.

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“…The initial motivation for these questions lays in the theory of Moufang sets and its connection to (twin) trees. (See [2] for an introduction into this topic.) Suppose that T is a tree and G a subgroup of AutT such that the following hold:…”

Section: Introductionmentioning

confidence: 99%

Multiplicative quadratic maps

Grüninger

2014

Preprint

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In this paper we prove that a multiplicative quadratic map between a unital ring K and a field L is induced by a homomorphism from K into L or a composition algebra over L. Especially we show that if K is a field, then every multiplicative quadratic map is the product of two field homomorphisms. Moreover, we prove a multiplicative version of Artin's Theorem showing that a product of field homomorphisms is unique up to multiplicity.

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Trees, contraction groups, and Moufang sets

Cited by 8 publications

References 32 publications

Locally Normal Subgroups of Totally Disconnected Groups. Part Ii: Compactly Generated Simple Groups

Locally Normal Subgroups of Totally Disconnected Groups. Part Ii: Compactly Generated Simple Groups

Locally compact convergence groups and $$n$$ n -transitive actions

Multiplicative quadratic maps

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