Existence of Lattices in Kac–moody Groups Over Finite Fields

Carbone, Lisa; Garland, Howard

doi:10.1142/s0219199703001117

Cited by 48 publications

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“…4 Recall that BN-pairs are also called Tits systems. 5 Recall that by 'subbase' for a topology on a group, we mean a 'subbase of neighborhoods of the identity'.…”

Section: Complete Kac-moody Groupsmentioning

confidence: 99%

“…Distinct constructions of complete Kac-Moody groups are given in the papers of Carbone and Garland [4] and Remy and Ronan [20]; in both cases the group constructed is the completion of Tits' group G(A) with respect to a certain topology. In [20], the topology comes from the action of G(A) on its associated positive building; we denote the corresponding completion by G(A).…”

Section: Introductionmentioning

confidence: 99%

“…In [20], the topology comes from the action of G(A) on its associated positive building; we denote the corresponding completion by G(A). 1 In [4], one starts with the integrable highest weight module V λ for the Kac-Moody algebra g (A) corresponding to a regular dominant integral weight λ, then considers a certain Z-form, V λ Z of V λ , and the action of G(A) on a k-form, V λ k of V λ Z . The corresponding completion of G (which depends on λ) will be denoted by G λ (A).…”

Section: Introductionmentioning

confidence: 99%

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Abstract simplicity of complete Kac–Moody groups over finite fields

Carbone

Ershov

Ritter

2008

Journal of Pure and Applied Algebra

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simplicity of complete Kac-Moody groups over finite fields Communicated by C.A. Weibel MSC:Primary: 20E42 secondary: 20E32 17B67 20E18 22F50 a b s t r a c t Let G be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix A, as constructed by Tits. It is known that G admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac-Moody group G which is defined to be the closure of G in the automorphism group of its building. Our main goal is to determine when complete Kac-Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of G was previously known to hold when A is of affine type. We extend this result to many indefinite cases, including all hyperbolic generalized Cartan matrices A of rank at least four. Our proof uses Tits' simplicity theorem for groups with a BN-pair and methods from the theory of pro-p groups.Remark. In the hypotheses of Theorem 1.1, a submatrix is not necessarily proper.Abstract simplicity of G(A) was previously known only when A is of finite type (in which case G(A) is a finite group) or A is of affine type, in which case G(A) is isomorphic to the group of K-points of a simple algebraic group defined over K = k((t))[18]. The class of matrices covered by Theorem 1.1 includes many indefinite examples, including all hyperbolic matrices of rank at least four -see Proposition 2.1. Remark. Recently, Caprace and Remy [3] proved (abstract) simplicity of the incomplete group G(A) modulo its finite centerin the case when the associated Coxeter group is not affine and assuming that |k| is sufficiently large. If A is affine, the incomplete group G(A) (modulo its center) is infinite and residually finite, and hence cannot be simple.Briefly, our approach to proving Theorem 1.1 will be as follows. A celebrated theorem of Tits [2] gives sufficient conditions (called "simplicity axioms") for a group with a BN-pair to be simple. The group G(A) routinely satisfies most of these axioms if |k| > 3 (for arbitrary A), which already implies topological simplicity of G(A). It is not clear if the remaining axioms. We have already defined the group G J . Note that by relations (R3), G J ⊃ U α for every α ∈ Φ J . Also introduce the subgroupsIn view of this proposition, we can identify U J (resp. G J ) with U(A J ) (resp. G(A J )). Let U J be the closure of U(A J ) in G(A J ), as before, and let U J be the closure of U J in G.Theorem 5.2. Assume that A is symmetric. The groups U J and U J are (topologically) isomorphic.Proof of Proposition 5.1. As before, identify Φ(A J ) with the subset Φ J of Φ. To distinguish between generators of G(A J ) and G(A), we use the symbols {x α (u) | α ∈ Φ(A J ), u ∈ k} for the generators of G(A J ) (the generators of G(A) are denoted {χ α (u)} as usual). From the defining presentation of Kac-Moody groups, it is clear that there exists a map ϕ :Clearly, ϕ(G(A J )) = G J , ϕ(U(A J )) = U J and ϕ(B(A J )) = B J , so we only need to show that ϕ is injective. We proceed in ...

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“…4 Recall that BN-pairs are also called Tits systems. 5 Recall that by 'subbase' for a topology on a group, we mean a 'subbase of neighborhoods of the identity'.…”

Section: Complete Kac-moody Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Abstract simplicity of complete Kac–Moody groups over finite fields

Carbone

Ershov

Ritter

2008

Journal of Pure and Applied Algebra

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“…Then ƒ has locally finite, regular right-angled twin buildings X C Š X , and ƒ acts diagonally on the product X C X . For q large enough: (a) By Theorem 0.2 of Carbone and Garland [4] or Theorem 1(i) of Rémy [11], the stabilizer in ƒ of a point in X is a non-cocompact lattice in Aut.X C /. Any such lattice is contained in a negative maximal spherical parabolic subgroup of ƒ, which has strict fundamental domain a sector in X C , and so any such lattice has strict fundamental domain.…”

Section: E42 20f05; 20f55 57m07 51e24mentioning

confidence: 99%

Infinite generation of non-cocompact lattices on right-angled buildings

Thomas

Wortman

2011

Algebr. Geom. Topol.

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“…For q large enough, a complete Kac-Moody group G over F q admits non-cocompact lattices, as established by Carbone-Garland [9] and independently Rémy [19]. In rank n = 2, where the building for G is a tree, various constructions of cocompact lattices in G are also known (see [5] and its references therein).…”

Section: Introductionmentioning

confidence: 97%

Co-compact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups

Capdeboscq

Thomas

2013

Mathematical Research Letters

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Abstract. Let G be a complete Kac-Moody group of rank n ≥ 2 over the finite field of order q, with Weyl group W and building Δ. We first show that if W is right-angled, then for all q ≡ 1 (mod 4) the group G admits a cocompact lattice Γ which acts transitively on the chambers of Δ. We also obtain a cocompact lattice for q ≡ 1 (mod 4) in the case that Δ is Bourdon's building. As a corollary of our constructions, for certain rightangled W and certain q, the lattice Γ has a surface subgroup. Our second main result states that if W is a free product of spherical special subgroups, then for all q, the group G admits a cocompact lattice Γ, with Γ a finitely generated free group. Thus for many G of rank n ≥ 3, our results provide the first (explicit) constructions of cocompact lattices in G, and in particular, the first explicit examples of cocompact lattices in complete Kac-Moody groups whose Weyl group is not right-angled. Our proofs use generalizations of our results in rank 2 [5] concerning the action of certain finite subgroups of G on Δ, together with covering theory for complexes of groups.

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Existence of Lattices in Kac–moody Groups Over Finite Fields

Cited by 48 publications

References 23 publications

Abstract simplicity of complete Kac–Moody groups over finite fields

Abstract simplicity of complete Kac–Moody groups over finite fields

Infinite generation of non-cocompact lattices on right-angled buildings

Co-compact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups

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