We consider nite lattice packings C n + K of n copies of some centrally symmetric convex body K in E d for large n. Assume that C n is a subset of a lattice , and % is at least the covering radius; namely, + %K covers the space. The parametric density (C n ; %) is deened by (C n ; %) = n V (K)=V (conv C n + %K). We show that if (C n ; %) is minimal for n large then the shape of conv C n is appr...

We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups.

Introduction. By a lattice homorphism of a group G onto a group G' we mean a single-valued mapping of the lattice L(G) of subgroups of G onto the lattice L(G') of subgroups of G', which preserves all unions and intersections, that is, which is subject to the conditions 1. (U,S,)0 = U,(S¿), 2. (r\vSr) = Ç)ASd>), for every (finite or infinite) set of subgroups 5„ of G. We call proper any l...

This note aims to clarify what conditions on a connected Lie group G imply that its maximal connected normal solvable subgroup R intersects each lattice of G as a lattice in R.

Let X be a locally finite tree and let G = Aut(X). Then G is naturally a locally compact group. A discrete subgroup Γ ≤ G is called an X-lattice, or a tree lattice if Γ has finite covolume in G. The lattice Γ is encoded in a graph of finite groups of finite volume. We describe several methods for constructing a pair of X-lattices (Γ′,Γ) with Γ ≤ Γ′, starting from ‘edgeindexed graphs’ (A′, i′) a...

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally...

For a left vector space V over a totally ordered division ring F, let Co(V ) denote the lattice of convex subsets of V . We prove that every lattice L can be embedded into Co(V ) for some left F-vector space V . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional, and L embeds into a finite lower bounded lattice of the form Co(V,Ω) = {X ∩Ω | X ∈ Co(V )}, for some f...

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally...

Let G and H ⊂ G be connected reductive real algebraic groups defined over Q, and admitting no nontrivial Q-characters. Let Γ ⊂ G(Q) be an arithmetic lattice in G, and π : G→ Γ\G be the natural quotient map. Let μH denote the H-invariant probability measure on the closed orbit π(H). Suppose that π(Z(H)) is compact, where Z(H) denotes the centralizer of H in G. We prove that the set {μH · g : g ∈...

We show that the degree of a graded lattice ideal of dimension 1 is the order of the torsion subgroup of the quotient group of the lattice. This gives an efficient method to compute the degree of this type of lattice ideals.