We study the lattice of finite-index extensions of a given finitely generated subgroup H of a free group F . This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of H . This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a sub-exp...

Suppose <? denotes a class of totally ordered groups closed under taking subgroups and quotients by o-homomorphisms. We study the following classes: (1) Res (£?), the class of all lattice-ordered groups which are subdirect products of groups in C; (2) Hyp(C), the class of lattice-ordered groups in Res(C) having all their ¿-homomorphic images in Res(<?); Para(C), the class of lattice-ordered gro...

We extend a model of community isolation in the d-dimensional lattice to a case with an imposed imbalance between the birth rates of competing communities. We provide analytical and numerical evidences that in the asymmetric two-species model there exists a well-defined value of the asymmetry parameter when the emergence of the isolated (blocked) subgroups is the fastest, i.e., the characterist...

Consider G = SLd(R) and Γ = SLd(Z). It was recently shown by the second-named author [21] that for some diagonal subgroups {gt} ⊂ G and unipotent subgroups U ⊂ G, gttrajectories of almost all points on all U -orbits on G/Γ are equidistributed with respect to continuous compactly supported functions φ on G/Γ. In this paper we strengthen this result in two directions: by exhibiting an error rate ...

Let G be a semi-simple Lie group, and Γ < G a lattice. Following Sarnak (see [19]), a finitely generated subgroup ∆ of Γ is called thin if ∆ has infinite index in Γ, but is Zariski dense. There has been a good deal of interest recently in thin groups (see for example [7], [8] and [19] to name a few), and there are many results that give credence to the statement that “generic subgroups of latti...

In 1993, Muzychuk [18] showed that the rational Schur rings over a cyclic group Zn are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Zn. This can easily be extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational Schur rings over G ...

Freese and J6nsson [8] showed that the congruence lattice of a (universal) algebra in a congruence modular variety is always arguesian. On the other hand J6nsson [16] constructed arguesian lattices which cannot be embedded into the normal subgroup lattice of a group. These lattices consist of two arguesian planes of different prime order glued together over a two element sublattice (cf. Dilwort...

Let W be a Coxeter group acting as a matrix group by way of the dual of the geometric representation. Let L be the lattice of intersections of all reflecting hyperplanes associated with the reflections in this representation. We show that L is isomorphic to the lattice consisting of all parabolic subgroups of W . We use this correspondence to find all W for which L is supersolvable. In particul...

The answer is yes, if ... This note attempts to give amplification to the above statement, while at the same time arriving at a reasonable description of this lattice. The main theorem of the paper is no doubt the assertion that the lattice of torsion classes of lattice-ordered groups is completely distributive. The proof of this theorem depends on the notion of a value selector, and should not...