Transformation Groups, Vol. 4, No. 4, 1999, pp. 329-353 C) Birkh~user Boston (1999) THE SIZE OF A HYPERBOLIC COXETER SIMPLEX N. W. JOHNSON Department of Mathematics and Computer Science Wheaton College Norton, MA 02766, U.S.A. njohnson@wheatonma.edu R. KELLERHALS Mathematisches Institut Universit~it GSttingen 37073 GSttingen, Germany ruth@cfgauss.uni-math.gwdg.de J. G. RATCLIFFE Department of Mathematics Vanderbilt University Nashville, TN 37240, U.S.A. ratclifj @math.vanderbilt.edu S. T. TSCHANTZ Department of Mathematics Vanderbilt University Nashville, TN 37240, U.S.A. tschantz@math.vanderbilt.edu Abstract. We determine the covolumes of all hyperbolic Coxeter simplex reflection groups. These groups exist up to dimension 9. The volume computations involve several different methods according to the parity of dimension, subgroup relations and arithmeticity properties. Introduction Let X" = S n, E n, or H n denote either spherical, Euclidean, or hyperbolic n-space. A Coxeter simplex is a n-dimensional simplex in X n, all of whose dihedral angles are submultiples of ~r or zero. We allow a simplex in H n to be unbounded with ideal vertices on the sphere at infinity OH n. A Coxeter simplex reflection group is a group generated by the reflections in the sides of a Coxeter simplex in X n. A Coxeter simplex reflection group is a discrete group of isometries of X '~, with fundamental domain its defining Co- xeter simplex. Spherical Coxeter simplex reflection groups are finite, whereas Euclidean and hyperbolic Coxeter simplex reflection groups are infinite. Coxeter simplex reflection groups arise naturally in geometry as groups of symmetries of regular tessellations of X n. The spherical and Euclidean Coxeter simplices were classified by H. S. M. Coxeter [C1]. The hyperbolic Coxeter simplices were classified by H. S. M. Coxeter and G. J. Whitrow [CW], F. Lann~r ILl, J.-L. Koszul [K], and M. Chein [C]. For each dimension n > 3, there are only finitely many hyperbolic Coxeter simplices, and such simplices exist only in dimensions n = 2, 3,..., 9. By the size of a non-Euclidean Coxeter simplex, we mean its n-dimensional volume in X n. For a spherical Coxeter simplex, this is just the volume of S n, 27r(n+l)/2 v~ - F((n + 1)/2)' Received March 27, 1998. Accepted July 24, 1998.