Lisa Carbone

Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits

We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin diagram of compact hyperbolic type is 4. Building on earlier classification results of Kac, Kobayashi-Morita, Li and Sacliolu, we present the 238 hyperbolic Dynkin diagrams in ranks 3–10, 142 of which are symmetrizable. For each symmetrizable hyperbolic generalized Cartan matrix, we give a symmetrization and hence the distinct lengths of real roots in the corresponding root system. For each such hyperbolic root system we determine the disjoint orbits of the action of the Weyl group on real roots. It follows that the maximal number of disjoint Weyl group orbits on real roots in a hyperbolic root system is 4.

Abstract simplicity of complete Kac-Moody groups over finite fields

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 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 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.

Non-Uniform Lattices on Uniform Trees

Consider a locally finite tree X, a subgroup \(H \leqslant G = Aut(X)\) without inversions, and indexed quotient graph \((A,i) = I(H\backslash \backslash X).\)

Groups acting simply transitively on vertex sets of hyperbolic triangular buildings

We construct and classify all groups, given by triangular presentations associated to the smallest thick generalized quadrangle, that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. Our classification shows 23 non-isomorphic torsion free groups (obtained in an earlier work) and 168 non-isomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the Euclidean case, we find both torsion and torsion free groups acting onWe construct and classify all groups acting simply transitively on vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. We have both torsion and torsion free groups acting on the same building.

Lattices on Nonuniform Trees

Let X be a locally finite tree, and let G = Aut(X). Then G is a locally compact group. We show that if X has more than one end, and if G contains a discrete subgroup Γ such that the quotient graph of groups Γ\\X is infinite but has finite covolume, then G contains a nonuniform lattice, that is, a discrete subgroup Λ such that Λ\G is not compact, yet has a finite G-invariant measure.

LATTICES ON PARABOLIC TREES

ABSTRACT Let X be a locally finite tree, and . Then G is a locally compact group. A non-uniform X-lattice is a discrete subgroup such that the quotient graph of groups is infinite but has finite covolume, and a non-uniform G-lattice is a discrete subgroup such that is not compact yet has a finite G-invariant measure. We show that if X has a unique end and if G contains a non-uniform X-lattice, then G contains a non-uniform G-lattice if and only if any path directed towards the end of the edge-indexed quotient of X has unbounded index.

INFINITE DESCENDING CHAINS OF COCOMPACT LATTICES IN KAC–MOODY GROUPS

Let A be a symmetrizable affine or hyperbolic generalized Cartan matrix. Let G be a locally compact Kac–Moody group associated to A over a finite field 𝔽q. We suppose that G has type ∞, that is, the Weyl group W of G is a free product of ℤ/2ℤs. This includes all locally compact Kac–Moody groups of rank 2 and three possible locally compact rank 3 Kac–Moody groups of noncompact hyperbolic type. For every prime power q, we give a sufficient condition for the rank 2 Kac–Moody group G to contain a cocompact lattice with quotient a simplex, and we show that this condition is satisfied when q = 2s. If further Mq and are abelian, we give a method for constructing an infinite descending chain of cocompact lattices … Γ3 ≤ Γ2 ≤ Γ1 ≤ Γ. This allows us to characterize each of the quotient graphs of groups Γi\\X, the presentations of the Γi and their covolumes, where X is the Tits building of G, a homogeneous tree. Our approach is to extend coverings of edge-indexed graphs to covering morphisms of graphs of groups with abelian groupings. This method is not specific to cocompact lattices in Kac–Moody groups and may be used to produce chains of subgroups acting on trees in a general setting. It follows that the lattices constructed in the rank 2 Kac–Moody group have the Haagerup property. When q = 2 and rank(G) = 3 we show that G contains a cocompact lattice Γ′1 that acts discretely and cocompactly on a simplicial tree . The tree is naturally embedded in the Tits building X of G, a rank 3 hyperbolic building. Moreover Γ′1 ≤ Λ′ for a non-discrete subgroup Λ′ ≤ G whose quotient Λ′ \ X is equal to G\X. Using the action of Γ′1 on we construct an infinite descending chain of cocompact lattices …Γ′3 ≤ Γ′2 ≤ Γ′1 in G. We also determine the quotient graphs of groups , the presentations of the Γ′i and their covolumes.

Non-minimal tree actions and the existence of non-uniform tree lattices

A uniform tree is a tree that covers a finite connected graph. Let X be any locally finite tree. Then G = Aut(X) is a locally compact group. We show that if X is uniform, and if the restriction of G to the unique minimal G-invariant subtree X0 ⊆ X is not discrete then G contains non-uniform lattices; that is, discrete subgroups Γ for which Γ\G is not compact, yet carries a finite G-invariant measure. This proves a conjecture of Bass and Lubotzky for the existence of non-uniform lattices on uniform trees.

The tree lattice existence theorems

Let X be a locally finite tree, and let G=Aut(X). Then G is a locally compact group. In analogy with Lie groups, Bass and Lubotzky conjectured that G contains lattices, that is, discrete subgroups whose quotient carries a finite invariant measure. Bass and Kulkarni showed that G contains uniform lattices if and only if G is unimodular and G⧹X is finite. We describe the necessary and sufficient conditions for G to contain lattices, both uniform and non-uniform, answering the Bass–Lubotzky conjectures in full. To cite this article: L. Carbone, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 223–228.

Integral group actions on symmetric spaces and discrete duality symmetries of supergravity theories

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