We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of PSL(2,C). We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between el...

Question 1.1. Let G be a connected Lie group and let A = (A1, . . . , Ak) be a finite ordered subset of G. Is the discreteness problem for the subgroup ΓA := 〈A1, . . . , Ak〉 < G decidable? This question, in the case of G = PSL(2,C), was raised, most recently, in the paper [8] by J. Gilman and L. Keen, who noted that “it is a challenging problem that has been investigated for more than a centur...

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dense subgroups with non-empty domain of discontinuity have discrete commensurato...

We introduce notions of finite presentation and co-exactness which serve as qualitative quantitative analogues finite-dimensionality for operator modules over completely contractive Banach algebras. With these we begin the development a local theory by introducing lifting property, nuclearity, semi-discreteness. For large class prove that property is equivalent to flatness, generalizing space r...

The main result is to show that the space of nonmonotonic fuzzy measures on a measurable space (X,X) with total variation norm is separable if and only if the -algebra X is a finite set. Our result is related to fuzzy analysis, functional spaces and discrete mathematics. © 2007 Elsevier B.V. All rights reserved.

Animal behaviors are sometimes decomposable into discrete, stereotyped elements. In one model, such behaviors are triggered by specific commands; in the extreme case, the discreteness of behavior is traced to the discreteness of action potentials in the individual command neurons. We use the crawling behavior of the nematode C. elegans to explore the opposite extreme, in which discreteness and ...