Bounded presentations of Kac–Moody groups

Capdeboscq, Inna

doi:10.1515/jgt-2013-0028

Cited by 5 publications

(10 citation statements)

References 6 publications

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“…Theorem 1.4(iii) of [2] gives the finite presentability of PSt A (R) if every pair of nodes of the Dynkin diagram lies in some irreducible spherical diagram of rank ≥ 3. (This use of a covering of A by spherical diagrams was also used by Capdeboscq [7].) By inspecting the list of affine Dynkin diagrams of rank > 3, one checks that this treats all cases of (i ) except A = α β γ δ (with some orientations of the double edges).…”

Section: Finite Presentationsmentioning

confidence: 99%

Presentation of affine Kac–Moody groups over rings

Allcock

2016

Algebra Number Theory

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Abstract. Tits has defined Steinberg groups and Kac-Moody groups for any root system and any commutative ring R. We establish a Curtis-Tits style presentation for the Steinberg group St of any rank ≥ 3 irreducible affine root system, for any R. Namely, St is the direct limit of the Steinberg groups coming from the 1-and 2-node subdiagrams of the Dynkin diagram. In fact we give a completely explicit presentation. Using this we show that St is finitely presented if the rank is ≥ 4 and R is finitely generated as a ring, or if the rank is 3 and R is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac-Moody groups when R is a Dedekind domain of arithmetic type.

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Section: Finite Presentationsmentioning

confidence: 99%

Presentation of affine Kac–Moody groups over rings

Allcock

2016

Algebra Number Theory

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“…Related results for other Kac-Moody groups over finite fields were also proved in [4]. As a consequence, the number of generators of a 2-spherical Kac-Moody group is independent of q and depends on the type of Dynkin diagram of G(q) rather than on the rank of G. We make use of this observation to provide bounds on the minimal number of generators of G(q).…”

Section: Introductionmentioning

confidence: 75%

“…In [1], Abramenko and Muhlherr have shown that with some restrictions (if the groups are 2-spherical, with some mild bounds on the size of F q ), Kac-Moody groups over F q are finitely presented with the number of generators depending on q and the Lie rank of G(q). 1 In [4], the author has shown that the family of affine Kac-Moody groups over F q (of rank at least 3) possesses bounded presentations: there exists C > 0 such that if G(q) is an affine Kac-Moody group of rank at least 3 corresponding to an indecomposable generalised Cartan matrix (IGCM) and q ≥ 4, then G(q) has a presentation with d(G) generators and r(G) relations satisfying d(…”

Section: Introductionmentioning

confidence: 99%

On the generation of discrete and topological Kac–Moody groups

Capdeboscq

2015

Comptes Rendus. Mathématique

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Presented by the Editorial BoardThis article shows that discrete or topological Kac-Moody groups defined over finite fields are in many cases 2-generated. We provide explicit bounds on the minimal number of generators for arbitrary Kac-Moody groups.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é On montre que les groupes de Kac-Moody topologiques ou discrets définis sur des corps finis sont 2-engendrés dans de nombreux cas. On exhibe ensuite des bornes explicites sur le nombre minimal de générateurs pour un groupe de Kac-Moody arbitraire.On considère des groupes de Kac-Moody sur des corps finis F q . Théorème 0.1. Soit G = G(q) un groupe de Kac-Moody simplement connexe de rang m correspondant à une matrice de Cartan généralisée indécomposable (MCGI) A, défini sur un corps fini F q , q = p a . Soit π = {α 1 , . . . , α m } l'ensemble des racines simples de G et soit le diagramme de Dynkin de G dont les sommets sont numérotées par α 1 , . . . , α m . Posons que, pour tout sous-ensembleminimal d'éléments de G nécessaires pour générer G. Alors lorsque q est suffisamment grand, on a :dans au moins 34 des 72 cas) à part peut-être dans trois cas exceptionnels de rang 3 et pour lesquels est de type (∞, ∞, ∞). Dans ces trois cas, d(G) ≤ 4 ; E-mail address: I.Capdeboscq@warwick.ac.uk. http://dx.(v) supposons que π peut être découpé en k sous-ensembles mutuellement disjoints π i , 1 ≤ i ≤ k, tels que π i = {α i 1 , . . . , α i l(i) } avec α i j ∈ π , 1 ≤ j ≤ l(i) (où l(i) = |π i |) et que pour chaque i ∈ {1, . . . , k − 1}, on a (π i ) = s(i) j=1 ij où ij est un diagramme de Dynkin irréductible de type fini (ce qui signifie que (π i ) peut être découpé en s(i) diagrammes de Dynkin de type fini où s(i) ∈ N dépend de π i ). Alors : (a) si (π k ) = r j=1 kj où kj est un diagramme de Dynkin irréductible de type fini, alors d(G) ≤ 2k ; (b) si (π k ) = r j=1 kj où kj est un diagramme de Dynkin irréductible de rang 2 de type infini, alors d(G) ≤ 2k + 2, et, si q est assez grand, d(G) ≤ 2k + 1. Exemple 1. Si est un arbre enraciné fini de profondeur m, d(G) ≤ 4 lorsque q ≥ √ m. Corollaire 0.2. Soit G un groupe de Kac-Moody minimal défini sur un corps F q , avec q = p a et p ≥ max i = j |a ij | (où A = (a ij ) est la MCGI de G). Soit G le groupe de Kac-Moody topologique correspondant à G. Alors les conclusions du Théorème 0.1 sont vraies si on remplace G par G et si d(G) représente le nombre minimal de générateurs topologiques de G.

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“…In recent years several papers have been dedicated to showing that finite simple groups or arithmetic groups have presentations with bounded number of generators and relations [KL06], [GKKL07], [GKKL08], [GKKL11], [Cap13]. This paper is about such quantitative results for some classes of profinite groups.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 0.1 is proved by combining the results of [Cap13] on bounded presentations of the discrete Kay-Moody group G(F q [t, t −1 ]), with the fact that it has the congruence subgroup property. It is important at this point to note that the above uniformness statement is not true globally, that is for abstract presentations of arithmetic groups obtained by replacing the local rings F q [[t]] by rings of integers of global fields (see §3 below).…”

Section: Introductionmentioning

confidence: 99%