This paper investigates
transitive groups of direct isometries, without fixed points, of hyperbolic n-space Hn.
For n = 2 there is a natural one-to-one correspondence between the set of all such
groups and the set of ideal points of H2. For n ≧ 3 there is an analogous collection
of groups, which are in several senses the simplest but not the only such
groups.
The existence of a transitive group of transformations without fixed points can be
used to define an addition of points in the transformed space. The idea of sums of
points in hyperbolic spaces has been used in probabilistic applications, for example
by Kifer and by Karpelevich, Tutubalin and Shur. These involve a composition of
measures based on the collection (not a group) of translations of H2 or H3. The
group structure seems necessary for certain statistical questions, such as
characterizations of normal distributions, which were in part the motivation for this
investigation.
