OPTi's Algorithm for Discreteness Determination

OPTi's Algorithm for Drawing the Limit Set

Informative Words and Discreteness

There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichmüller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of such words and sequences: the semigroup of so called good words of Gehring-Martin...

Complexity and Discreteness Criteria

Let G be a subgroup of PSL(2, C). The discreteness problem for G is the problem of determining whether or not G is discrete. In this paper we assume that all groups considered are non-elementary. If G is generated by two elements, A and B of PSL(2, C), we have what is called the two-generator discreteness problem. Two-generator groups are important because, by a result of Jorgensen, an arbitrar...

Canonical Hexagons and the Psl(2,c) Discreteness Problem

The discreteness problem, that is, the problem of determining whether or not a given finitely generated group G of orientation preserving isometries of hyperbolic three-space, H is discrete as a subgroup of the whole isometry group Isom(H), is a challenging problem that has been investigated for more than a century and is still open. It is known that G is discrete if, and only if, every non-ele...

Discreteness Issues in Actuator Arrays

An actuator array is a form of distributed manipulation where an object being transported and manipulated rests on a large number of stationary supporting actuators. The authors have developed a macroscopic actuator array consisting of many motorized wheels. The analysis of such an array as opposed to a MEMS array requires the explicit modeling of the discreteness in the system, including the s...