Lattice Of Rank Research Articles

Congruence Subgroups of Lattices in Rank 2 Kac–Moody Groups over Finite Fields

Let G be a rank 2 complete affine or hyperbolic simply-connected Kac–Moody group over a finite field k. Then G is locally compact and totally disconnected. Let B− = HU− be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U− is generated by all negative real root groups. Let w1 and w2 be the generators of the Weyl group. Let be the negative standard parabolic subgroup of G corresponding to w1. It is known that the subgroups U−, B− and , are nonuniform lattice subgroups of G. Here we construct an infinite sequence of congruence subgroups of as natural generalizations of the corresponding notions for lattices in Lie groups. We also show that the group U− contains analogous congruence subgroups. Our technique involves determining graphs of groups presentations for U−, B−, and with the fundamental apartment of the Bruhat–Tits tree X a quotient graph for U− and for B− on X. When k = 𝔽q and q = 2s, the graph of groups for has the the positive half of the fundamental apartment as quotient graph. We explicitly construct the graphs of groups for the principal (level 1) congruence subgroup of and the analogous subgroups of U− giving generalized amalgam presentations for them.

IMAGES OF QUANTUM REPRESENTATIONS OF MAPPING CLASS GROUPS AND DUPONT–GUICHARDET–WIGNER QUASI-HOMOMORPHISMS

We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further, we show that the images of the mapping class groups have non-trivial 2-cohomology, at least for small levels. For this purpose, we considered a series of quasi-homomorphisms on mapping class groups extending the previous work of Barge and Ghys (Math. Ann. 294(1992), 235–265) and of Gambaudo and Ghys (Bull. Soc. Math. France 133(4) (2005), 541–579). These quasi-homomorphisms are pull-backs of the Dupont–Guichardet–Wigner quasi-homomorphisms on pseudo-unitary groups along quantum representations.

Non-divergence of unipotent flows on quotients of rank-one semisimple groups

Let$G$be a semisimple Lie group of rank one and$\unicode[STIX]{x1D6E4}$be a torsion-free discrete subgroup of$G$. We show that in$G/\unicode[STIX]{x1D6E4}$, given$\unicode[STIX]{x1D716}>0$, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than$\unicode[STIX]{x1D6FF}$for a$1-\unicode[STIX]{x1D716}$proportion of the time, for some$\unicode[STIX]{x1D6FF}>0$. The result also holds for any finitely generated discrete subgroup$\unicode[STIX]{x1D6E4}$and this generalizes Dani’s quantitative non-divergence theorem [On orbits of unipotent flows on homogeneous spaces.Ergod. Th. & Dynam. Sys.4(1) (1984), 25–34] for lattices of rank-one semisimple groups. Furthermore, for a fixed$\unicode[STIX]{x1D716}>0$, there exists an injectivity radius$\unicode[STIX]{x1D6FF}$such that, for any unipotent trajectory$\{u_{t}g\unicode[STIX]{x1D6E4}\}_{t\in [0,T]}$, either it spends at least a$1-\unicode[STIX]{x1D716}$proportion of the time in the set with injectivity radius larger than$\unicode[STIX]{x1D6FF}$, for all large$T>0$, or there exists a$\{u_{t}\}_{t\in \mathbb{R}}$-normalized abelian subgroup$L$of$G$which intersects$g\unicode[STIX]{x1D6E4}g^{-1}$in a small covolume lattice. We also extend these results to when$G$is the product of rank-one semisimple groups and$\unicode[STIX]{x1D6E4}$a discrete subgroup of$G$whose projection onto each non-trivial factor is torsion free.

Indecomposable quadratic lattices over global function fields

In this paper, we prove that there exists an indecomposable lattice of rank 5 over a Hasse domain of any rational function field in which −1 is not a square, which solves a problem proposed by Gerstein. We also construct three concrete indecomposable lattices of rank 6 and rank 5 over a Hasse domain of the rational function field F7(x).

Approximation properties for noncommutative Lp -spaces of high rank lattices and nonembeddability of expanders

Abstract This article contains two rigidity type results for SL ⁢ ( n , ℤ ) {\mathrm{SL}(n,\mathbb{Z})} for large n that share the same proof. Firstly, we prove that for every p ∈ [ 1 , ∞ ] {p\in[1,\infty]} different from 2, the noncommutative L p {L^{p}} -space associated with SL ⁢ ( n , ℤ ) {\mathrm{SL}(n,\mathbb{Z})} does not have the completely bounded approximation property for sufficiently large n depending on p. The second result concerns the coarse embeddability of expander families constructed from SL ⁢ ( n , ℤ ) {\mathrm{SL}(n,\mathbb{Z})} . Let X be a Banach space and suppose that there exist β < 1 2 {\beta<\frac{1}{2}} and C > 0 {C>0} such that the Banach–Mazur distance to a Hilbert space of all k-dimensional subspaces of X is bounded above by C ⁢ k β {Ck^{\beta}} . Then the expander family constructed from SL ⁢ ( n , ℤ ) {\mathrm{SL}(n,\mathbb{Z})} does not coarsely embed into X for sufficiently large n depending on X. More generally, we prove that both results hold for lattices in connected simple real Lie groups with sufficiently high real rank.

Incompressible Surfaces in Rank One Locally Symmetric Spaces

We show that cocompact lattices in rank one simple Lie groups of non-compact type distinct from SO(2m, 1)(m ≥ 1) contain surface subgroups.

Spinor norm for skew-hermitian forms over quaternion algebras

We complete all local spinor norm computations for quaternionic skew-hermitian forms over the field Q of rational numbers. This can be used to compute the number of classes in a genus of skew-hermitian lattices of rank 2 or larger over a maximal order in a quaternion algebra D over Q in many cases, e.g., when D ramifies at infinity. Examples are provided.

A normal subgroup theorem for commensurators of lattices

We establish a general normal subgroup theorem for commensurators of lattices in locally compact groups. While the statement is completely elementary, its proof, which rests on the original strategy ofMargulis in the case of higher rank lattices, relies heavily on analytic tools pertaining to amenability and Kazhdan’s property (T). It is a counterpart to the normal subgroup theorem for irreducible lattices of Bader and the second named author, and may also be used to sharpen that result when one of the ambient factors is totally disconnected.

Heights of Kudla–Rapoport divisors and derivatives of $$L$$ L -functions

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight 2-n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of f. Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the central derivative of the convolution L-function of a weight n cusp form (depending on f) and the theta function of a positive definite hermitian lattice of rank n-1. When specialized to the case n=2, this result can be viewed as a variant of the Gross-Zagier formula for Shimura curves associated to unitary groups of signature (1,1). The proof relies on, among other things, a new method for computing improper arithmetic intersections.

Symmetry Groups Associated with Tilings of a Flat Torus

A flat torus E^2/Λ is the quotient of the Euclidean plane E^2 with a full rank lattice Λ generated by two linearly independent vectors v_1 and v_2. A motif-transitive tiling T of the plane whose symmetry group G contains translations with vectors v_1 and v_2 induces a tiling T^* of the flat torus. Using a sequence of injective maps, we realize T^* as a tiling T-of a round torus (the surface of a doughnut) in the Euclidean space E^3. This realization is obtained by embedding T^* into the Clifford torus S^1 × S^1 ⊆ E^4 and then stereographically projecting its image to E^3. We then associate two groups of isometries with the tiling T^* – the symmetry group G^* of T^* itself and the symmetry group G-of its Euclidean realization T-. This work provides a method to compute for G^* and G-using results from the theory of space forms, abstract polytopes, and transformation geometry. Furthermore, we present results on the color symmetry properties of the toroidal tiling T^* in relation with the color symmetry properties of the planar tiling T. As an application, we construct toroidal polyhedra from T-and use these geometric structures to model carbon nanotori and their structural analogs.

Umbral moonshine and the Niemeier lattices

In this paper, we relate umbral moonshine to the Niemeier lattices - the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice, we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups, we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to the umbral moonshine conjecture, stating that an infinite-dimensional module is assigned to each of the Niemeier lattices in such a way that the associated graded trace functions are mock modular forms of a distinguished nature. These constructions and conjectures extend those of our earlier paper and in particular include the Mathieu moonshine observed by Eguchi, Ooguri and Tachikawa as a special case. Our analysis also highlights a correspondence between genus zero groups and Niemeier lattices. As a part of this relation, we recognise the Coxeter numbers of Niemeier root systems with a type A component as exactly those levels for which the corresponding classical modular curve has genus zero.

The Minimal System of Defining Relations of the Free Modular Lattice of Rank 3 and Lattices Close to Modular One

We construct the system of 11 defining relations for the 3-generated free modular lattice. This system is proved to be minimal. Systems of defining relations for lattices close to modular one are studied.

Defining relations of a free modular lattice of rank 3

For a 3-generated free modular lattice we obtain a set of 11 defining relations and prove that this set is minimal.

Lattices in finite fields

In this paper, we consider lattices in \(\overline{\mathbb{F }_q}\), a fixed algebraic closure of \(\mathbb{F }_q\). The objective of this paper is to introduce \(J\)-invariants for a lattice of arbitrary rank. Theses \(J\)-invariants classify lattices by isomorphism. Next, we define singular lattices that are similar to classical lattices with complex multiplication. Then, we establish a result for the modular field, which is generated by the values of the \(J\)-invariants of a given singular lattice.

On the Classification of Lattices Over Which Are Even Unimodular -Lattices of Rank 32

We classify the lattices of rank 16 over the Eisenstein integers which are even unimodular -lattices (of dimension 32). There are exactly 80 unitary isometry classes.

Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau–Ginzburg model coming from a toric degeneration.

Classification of simple Lie algebras on a lattice

Let $\Lambda$ be a lattice of rank $n$. A Lie algebra on the lattice $\Lambda$ is a Lie algebra ${\cal L}=\oplus_{\lambda\in\Lambda}\,{\cal L}_{\lambda}$ such that $\dim\,{\cal L}_\lambda=1$ for all $\lambda$. In this article, we classify all simple graded Lie algebras on a lattice.

ON THE CLASSIFICATION OF EVEN UNIMODULAR LATTICES WITH A COMPLEX STRUCTURE

This paper classifies the even unimodular lattices that have a structure as a Hermitian [Formula: see text]-lattice of rank r ≤ 12 for rings of integers in imaginary quadratic number fields K of class number 1. The Hermitian theta series of such a lattice is a Hermitian modular form of weight r for the full modular group, therefore we call them theta lattices. For arbitrary imaginary quadratic fields we derive a mass formula for the principal genus of theta lattices which is applied to show completeness of the classifications.

Fixed point property for universal lattice on Schatten classes

The special linear group G = SLn(Z[x(1), ... , x(k)]) (n at least 3 and k finite) is called the universal lattice. Let n be at least 4, and p be any real number in (1, infinity). The main result is the following: any finite index subgroup of G has the fixed point property with respect to every affine isometric action on the space of p-Schatten class operators. It is in addition shown that higher rank lattices have the same property. These results are a generalization of previous theorems respectively of the author and of Bader-Furman-Gelander-Monod, which treated a commutative L-p-setting.

Formula omitted]-Whitney numbers of Dowling lattices

Let G be a finite group of order m≥1. A Dowling lattice Qn(G) is the geometric lattice of rank n over G. In this paper, we define the r-Whitney numbers of the first and second kind over Qn(G), respectively. This concept is a common generalization of the Whitney numbers and the r-Stirling numbers of both kinds. We give their combinatorial interpretations over the Dowling lattice and we obtain various new algebraic identities. In addition, we develop the r-Whitney–Lah numbers and the r-Dowling polynomials associated with the Dowling lattice.