Vol. 92, No. 1, 1981

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Transitive groups of isometries on Hn

Esther Portnoy

Vol. 92 (1981), No. 1, 127–139

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.

Mathematical Subject Classification 2000
Primary: 53A35
Secondary: 53C30
Received: 16 October 1978
Published: 1 January 1981
Esther Portnoy