We showed in [Oh] that for a simple real Lie group G with real rank at least 2, if a discrete subgroup Γ of G contains lattices in two opposite horospherical subgroups, then Γ must be a non-uniform arithmetic lattice in G, under some additional assumptions on the horospherical subgroups. Somewhat surprisingly, a similar result is true even if we only assume that Γ contains a lattice in one horo...

We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at s = 1 with a real pole of order 2, improving upon a result of [11]. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less or equal N is O(N...

In this short note, we show that the class of abelian groups determined by the subgroup lattice of their direct n-powers is exactly the class of the abelian groups which share the n-root property. As applications we answer in the negative a (semi)conjecture of Palfy and solve a more general problem. Recently, for an arbitrary group G, the subgroup lattice of the square G×G has received some att...

The structure of the coincidence symmetry group of an arbitrary n-dimensional lattice in the n-dimensional Euclidean space is considered by describing a set of generators. Particular attention is given to the coincidence isometry subgroup (the subgroup formed by those coincidence symmetries that are elements of the orthogonal group). Conditions under which the coincidence isometry group can be ...

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 non-uniform lattice, that is, a discrete subgroup Λ such that Λ\G is not compact, yet has a finite G-invariant measure. 0. Notatio...

In abstract algebra courses, teachers are often confronted with the task of drawing subgroup lattices. For purposes of instruction, it is usually desirable that these lattices be planar graphs (with no crossings). We present a characterization of abelian groups with this property. We also resolve the following problem in the abelian case: if the subgroup lattice is required to be drawn hierarch...

1.1. Motivation and overview. Let G denote the identity component of the special orthogonal group SO(n, 1), n ≥ 2, and V a finite-dimensional real vector space on which G acts linearly from the right. A discrete subgroup of a locally compact group with finite covolume is called a lattice. For v ∈ V and a subgroup H of G, let Hv = {h ∈ H : vh = v} denote the stabilizer of v in H. A subgroup H of...

Let X be a complete simplicial toric variety over a finite field Fq with homogeneous coordinate ring S = Fq[x1, . . . , xr] and split torus TX ∼= (Fq). We prove that vanishing ideal of a subset Y of the torus TX is a lattice ideal if and only if Y is a subgroup. We show that these subgroups are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determin...