Lattice Of Rank Research Articles

Pattern rigidity in hyperbolic spaces: duality and PD subgroups

body { counter-reset: list; } ol li { list-style: none; } ol li:before { content: "("counter(list, decimal) ") "; counter-increment: list; } For i= 1,2 , let G_i be cocompact groups of isometries of hyperbolic space \mathbf{H}^n of real dimension n , n \geq 3 . Let H_i \subset G_i be infinite index quasiconvex subgroups satisfying one of the following conditions: We prove pattern rigidity for such pairs extending work of Schwartz who proved pattern rigidity when H_i is cyclic. All this generalizes to quasiconvex subgroups of uniform lattices in rank one symmetric spaces satisfying one of the conditions (1)–(4), as well as certain special subgroups with disconnected limit sets. In particular, pattern rigidity holds for all quasiconvex subgroups of hyperbolic 3-manifolds that are not virtually free. Combining this with results of Mosher, Sageev, and Whyte, we obtain quasi-isometric rigidity results for graphs of groups where the vertex groups are uniform lattices in rank one symmetric spaces and the edge groups are of any of the above types.

ON WELL-ROUNDED IDEAL LATTICES

We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study the well-rounded lattices coming from ideals in quadratic rings of integers, showing that there exist infinitely many real and imaginary quadratic number fields containing ideals which give rise to well-rounded lattices in the plane.

A joinings classification and a special case of Raghunathan’s conjecture in positive characteristic (with an appendix by Kevin Wortman)

We prove the classification of joinings for maximal horospherical subgroups acting on homogeneous spaces without any restriction on the characteristic. Using the linearization technique, we deduce a special case of Raghunathan’s orbit closure conjecture. In the appendix, quasi-isometries of higher rank lattices in semisimple algebraic groups over fields of positive characteristic are characterized.

On the Flag f-vector of a Graded Lattice with Nontrivial Homology

It is proved that the Boolean algebra of rank n minimizes the flag f-vector among all graded lattices of rank n, whose proper part has nontrivial top-dimensional homology. The analogous statement for the flag h-vector is conjectured in the Cohen-Macaulay case.

Lattices in complete rank 2 Kac–Moody groups

Let Λ be a minimal Kac–Moody group of rank 2 defined over the finite field Fq, where q=pa with p prime. Let G be the topological Kac–Moody group obtained by completing Λ. An example is G=SL2(K), where K is the field of formal Laurent series over Fq. The group G acts on its Bruhat–Tits building X, a tree, with quotient a single edge. We construct new examples of cocompact lattices in G, many of them edge-transitive. We then show that if cocompact lattices in G do not contain p-elements, the lattices we construct are the only edge-transitive lattices in G, and that our constructions include the cocompact lattice of minimal covolume in G. We also observe that, with an additional assumption on p-elements in G, the arguments of Lubotzky (1990) [21] for the case G=SL2(K) may be generalised to show that there is a positive lower bound on the covolumes of all lattices in G, and that this minimum is realised by a non-cocompact lattice, a maximal parabolic subgroup of Λ.

Greedy algorithm computing minkowski reduced lattice bases with quadratic bit complexity of input vectors

The authors present an algorithm which is a modification of the Nguyen-Stehle greedy reduction algorithm due to Nguyen and Stehle in 2009. This algorithm can be used to compute the Minkowski reduced lattice bases for arbitrary rank lattices with quadratic bit complexity on the size of the input vectors. The total bit complexity of the algorithm is \(O(n^2 \cdot (4n!)^n \cdot (\tfrac{{n!}} {{2^n }})^{\tfrac{n} {2}} \cdot (\tfrac{4} {3})^{\tfrac{{n(n - 1)}} {4}} \cdot (\tfrac{3} {2})^{\tfrac{{n^2 (n - 1)}} {2}} \cdot \log ^2 A) \), where n is the rank of the lattice and A is maximal norm of the input base vectors. This is an O(log2A) algorithm which can be used to compute Minkowski reduced bases for the fixed rank lattices. A time complexity n! · 3n(log A)O(1) algorithm which can be used to compute the successive minima with the help of the dual Hermite-Korkin-Zolotarev base was given by Blomer in 2000 and improved to the time complexity n! · (log A)O(1) by Micciancio in 2008. The algorithm in this paper is more suitable for computing the Minkowski reduced bases of low rank lattices with very large base vector sizes.

Lattice triangulations of $\mathbb{E}^3 $ and of the 3-torus

This paper gives answers to a few questions concerning tilings of Euclidean spaces where the tiles are topological simplices with curvilinear edges. We investigate lattice triangulations of Euclidean 3-space in the sense that the vertices form a lattice of rank 3 and such that the triangulation is invariant under all translations of that lattice. This is the dual concept of a primitive lattice tiling where the tiles are not assumed to be Euclidean polyhedra but only topological polyhedra. In 3-space there is a unique standard lattice triangulation by Euclidean tetrahedra (and with straight edges) but there are infinitely many non-standard lattice triangulations where the tetrahedra necessarily have certain curvilinear edges. From the view-point of Discrete Differential Geometry this tells us that there are such triangulations of 3-space which do not carry any flat discrete metric which is equivariant under the lattice. Furthermore, we investigate lattice triangulations of the 3-dimensional torus as quotients by a sublattice. The standard triangulation admits such quotients with any number n ≥ 15 of vertices. The unique one with 15 vertices is neighborly, i.e., any two vertices are joined by an edge. It turns out that for any odd n ≥ 17 there is an n-vertex neighborly triangulation of the 3-torus as a quotient of a certain non-standard lattice triangulation. Combinatorially, one can obtain these neighborly 3-tori as slight modifications of the boundary complexes of the cyclic 4-polytopes. As a kind of combinatorial surgery, this is an interesting construction by itself.

On Frobenius incidence varieties of linear subspaces over finite fields

We define Frobenius incidence varieties by means of the incidence relation of Frobenius images of linear subspaces in a fixed vector space over a finite field, and investigate their properties such as supersingularity, Betti numbers and unirationality. These varieties are variants of the Deligne–Lusztig varieties. We then study the lattices associated with algebraic cycles on them. We obtain a positive-definite lattice of rank 84 that yields a dense sphere packing from a 4-dimensional Frobenius incidence variety in characteristic 2.

Bohr and Besicovitch almost periodic discrete sets and quasicrystals

A discrete set $A$ in the Euclidian space is almost periodic if the measure with the unite masses at points of the set is almost periodic in the weak sense. We investigate properties of such sets in the case when $A-A$ is discrete. In particular, if $A$ is a Bohr almost periodic set, we prove that $A$ is a union of a finite number of translates of a certain full--rank lattice. If $A$ is a Besicovitch almost periodic set, then there exists a full-rank lattice such that in most cases a nonempty intersection of its translate with $A$ is large.

Groups with infinite mod- p Schur multiplier

The non-finiteness of the mod- p Schur multiplier of a finitely generated group G with trivial center implies the existence of uncountably many non-residually finite, non-isomorphic central extension groups with kernel C p (see Thm. A). This phenomenon is related to the comparison of the cohomology of the profinite completion G ˆ of a group G and the cohomology of the group G itself (see Thm. B); according to J-P. Serre [8] a group G is good if the cohomologies of G ˆ and G are naturally isomorphic on finite coefficients. It is shown that non-uniform arithmetic lattices of algebraic rank 1 groups over local fields of positive characteristic p are not good (see Thm. C).

The Conway–Sloane tetralattice pairs are non-isometric

Conway and Sloane constructed a 4-parameter family of pairs of isospectral lattices of rank four. They conjectured that all pairs in their family are non-isometric, whenever the parameters are pairwise different, and verified this for classical integral lattices of determinant up to 10 4. In this paper, we use our theory of lattice invariants to prove this conjecture.

A moonshine path for 5 A and associated lattices of ranks 8 and 16

We continue the program, begun in Griess and Lam (in press) [18], to make a moonshine path between a node of the extended E 8 -diagram and the Monster. Our goal is to provide a context for observations of McKay, Glauberman and Norton by realizing their theories in a more concrete form. In this article, we treat the 5 A-node. Most work in this article is a study of certain lattices of ranks 8, 16 and 24 whose determinants are powers of 5.

Bounds for solid angles of lattices of rank three

We find sharp absolute constants C 1 and C 2 with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [ C 1 , C 2 ] . In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in R N with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in R 3 . Such spherical configurations come up in connection with the kissing number problem.

The modularity of K3 surfaces with non-symplectic group actions

We consider complex K3 surfaces with a non-symplectic group acting trivially on the algebraic cycles. Vorontsov and Kondō classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this note is to study the Galois representations associated to these K3 surfaces. The rank of the transcendental lattices is even and varies from 2 to 20, excluding 8 and 14. We show that these K3 surfaces are dominated by Fermat surfaces, and hence they are all of CM type. We will establish the modularity of the Galois representations associated to them. Also we discuss mirror symmetry for these K3 surfaces in the sense of Dolgachev, and show that a mirror K3 surface exists with one exception.

Refined configuration results for extremal {Type II} lattices of ranks $40$ and $80$

We show that, if $L$ is an extremal Type II lattice of rank $40$ or $80$, then $L$ is generated by its vectors of norm $\operatorname {min}(L)+2$. This sharpens earlier results of Ozeki, and the second author and Abel, which showed that such lattices $L$ are generated by their vectors of norms $\operatorname {min}(L)$ and $\operatorname {min}(L)+2$.

Separation of relatively quasiconvex subgroups

Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF for closed hyperbolic 3-manifolds. The method is to reduce, via combination and filling theorems, the separability of a quasiconvex subgroup of a relatively hyperbolic group G to the separability of a quasiconvex subgroup of a hyperbolic quotient G/N. A result of Agol, Groves, and Manning is then applied.

Modular forms of weight 8 for Γ g (1, 2)

We complete the program indicated by the Ansatz of D’Hoker and Phong in genus 4 by proving the uniqueness of the restriction to Jacobians of the weight 8 Siegel cusp forms satisfying the Ansatz. We prove dim[Γ4(1, 2), 8]0 = 2 and dim[Γ4(1, 2), 8] = 7. In each genus, we classify the linear relations among the self-dual lattices of rank 16. We extend the program to genus 5 by constructing the unique linear combination of theta series that satisfies the Ansatz.

Minimally Supported Frequency Composite Dilation Wavelets

A composite dilation wavelet is a collection of functions generating an orthonormal basis for L2(ℝn) under the actions of translations from a full rank lattice and dilations by products of elements of non-commuting groups A and B. A minimally supported frequency composite dilation wavelet has generating functions whose Fourier transforms are characteristic functions of a lattice tiling set. In this paper, we study the case where A is the group of integer powers of some expanding matrix while B is a finite subgroup of the invertible n×n matrices. This paper establishes that with any finite group B together with almost any full rank lattice, one can generate a minimally supported frequency composite dilation wavelet system. The paper proceeds by demonstrating the ability to find such minimally supported frequency composite dilation wavelets with a single generator.

CONFIGURATIONS OF EXTREMAL EVEN UNIMODULAR LATTICES

We extend the results of Ozeki on the configurations of extremal even unimodular lattices. Specifically, we show that if L is such a lattice of rank 56, 72, or 96, then L is generated by its minimal-norm vectors.

Hermitian modular forms congruent to 1 modulo p

For any natural number $$\ell$$ and any prime $$p \equiv 1$$ (mod 4) not dividing $$\ell$$ there is a Hermitian modular form of arbitrary genus n over $$L := {\mathbb{Q}}[\sqrt{-\ell}]$$ that is congruent to 1 modulo p which is a Hermitian theta series of an O L -lattice of rank p − 1 admitting a fixed point free automorphism of order p. It is shown that also for non-free lattices such theta series are modular forms.