Hyperbolic Coxeter Groups
by
Norman W. Johnson
Wheaton College, Norton, Massachusetts 02766

Discrete groups generated by reflections in hyperbolic space, or hyperbolic Coxeter groups, have been investigated by Lanne'r, Coxeter and Whitrow, Koszul, Vinberg, and others. Klein's representations of linear fractional transformations over the rational or Gaussian integers as isometries of the hyperbolic plane or 3-space can be extended to other cases. Real, complex, and quaternionic modular groups are isomorphic to subgroups of certain hyperbolic Coxeter groups operating in spaces of dimension 2, 3, 4, or 5. Compact and paracompact groups, generated by reflections in the facets of ordinary or asymptotic simplexes, exist only in spaces of dimension less than 10. The sizes of the fundamental regions for all such groups have been calculated. Commensurability classes and subgroup relationships among them have also been determined.

{Note: The name "Lanne'r" has an "e" with an acute accent.}

Date received: March 26, 2004

Copyright © 2004 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # canm-20.