 
Summary: Reflection Groups on the Octave Hyperbolic Plane
Daniel Allcock*
4 August 1997
allcock@math.utah.edu
Department of Mathematics
University of Utah
Salt Lake City, UT 84112.
1991 mathematics subject classification: 22E40; secondary: 11F06, 17C40, 53C35.
Abstract.
For two different integral forms K of the exceptional Jordan algebra we show that AutK is gener
ated by octave reflections. These provide `geometric' examples of discrete reflection groups acting
with finite covolume on the octave (or Cayley) hyperbolic plane OH 2 , the exceptional rank one
symmetric space. (The isometry group of the plane is the exceptional Lie group F 4(\Gamma20) .) Our
groups are defined in terms of Coxeter's discrete subring K of the nonassociative division algebra
O and we interpret them as the symmetry groups of ``Lorentzian lattices'' over K. We also show
that the reflection group of the ``hyperbolic cell'' over K is the rotation subgroup of a particular
real reflection group acting on H 8 ¸ = OH 1 . Part of our approach is the treatment of the Jordan
algebra of matrices that are Hermitian with respect to any real symmetric matrix.
1 Introduction
The octave hyperbolic plane OH 2 is the exceptional rank one symmetric space; it is very similar to
