Let Γ be a lattice in G = SL(n,R) and X = G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of Γ in X analogous to the classical equidistribution on torus. To obtain this result, we first prove an ergodic theorem along balls in the connected component of Borel subgroup ...

An important and long-standing open problem in universal algebra asks whether every finite lattice is isomorphic to the congruence lattice of a finite algebra. Until this problem is resolved, our understanding of finite algebras is incomplete, since, given an arbitrary finite algebra, we cannot say whether there are any restrictions on the shape of its congruence lattice. If we find a finite la...

The aim of this paper is to give some connections between the structure of an abelian finite group and the structure of its subgroup lattice. Let (G, +) be an abelian group. Then the set L(G) of subgroups of G is a modular and complete lattice. Moreover, we suppose that G is finite of order n. If L n is the divisors lattice of n, then the following function is well defined: ord : L(G) −→ L n , ...

In the first Lecture, we saw that lattices can be viewed equivalently in two different ways, i.e. as discrete additive subgroup of R n or as an additive subgroup of R n with linearly independent generators. In this section, we show a final equivalent viewpoint which relates the discreteness of a lattice to the existence of an appropriate dual.

Based on the ideas in some recently uncovered notes of Selberg [14] on discrete subgroups of a product of SL2(R)’s, we show that a discrete subgroup of SL3(R) generated by lattices in upper and lower triangular subgroups is an arithmetic subgroup and hence a lattice in SL3(R).

Let G be a connected semisimple real algebraic group. Assume that G(R) has no compact factors and let Γ be a torsionfree uniform lattice subgroup of G(R). Then Γ contains a malnormal abelian subgroup A. This implies that the II1 factor VN(Γ) contains a masa A with Pukánszky invariant {∞}.

We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence latt...

Let G be a connected reductive group. To any irreducible G-variety X one assigns the lattice generated by all weights of B-semiinvariant rational functions on X , where B is a Borel subgroup of G. This lattice is called the weight lattice of X . We establish algorithms for computing weight lattices for homogeneous spaces and affine homogeneous vector bundles. For affine homogeneous spaces of ra...

The goal of this note is to give an elementary characterization of the well-known quaternion group Q8 by using its subgroup lattice.

It is shown that Aut(L ~ is naturally isomorphic to Aut(L) x Aut(Q) when L is a directly and exponentially indecomposable lattice, Q a non-empty connected poset, and one of the following holds: Q is arbitrary but L is a jm-lattice, Q is finitely factorable and L is complete with a join-dense subset of completely join-irreducible elements, or L is arbitrary but Q is finite. A problem of J6nsson ...