 
Summary: ON VOLUMES OF HYPERBOLIC ORBIFOLDS
ILESANMI ADEBOYE AND GUOFANG WEI
Abstract. In this paper we derive an explicit lower bound on the volume of a hyperbolic
norbifold for dimensions greater than or equal to four. Our main tool is H. C. Wang's
bound on the radius of a ball embedded in the fundamental domain of a discrete subgroup
of a semisimple Lie group.
0. Introduction
A hyperbolic norbifold Q is the quotient Hn/, where Hn denotes hyperbolic nspace
and represents a discrete group of orientationpreserving isometries of Hn. When con
tains no elements of finite order, Q is a hyperbolic nmanifold. The problem of determining
the hyperbolic norbifold and nmanifold of minimum volume has been an area of interest
for some time. While the solution in dimension 2 has been known since the nineteenth
century [11], the equivalent results for dimension 3 are very recent [6], [8]. Also of note
is the solution when is arithmetically defined for 3manifolds [4] and 3orbifolds [5]. In
dimension 4, the problem has been solved for the noncocompact [10] and arithmetic [2]
cases.
In [13], explicit lower bounds for the volume of hyperbolic manifolds in all dimensions
were constructed. The key was a generalization of Jørgensen's inequality [14] that provided
a link between the discreteness of and the geometry of Hn. An earlier paper [1], applied
these techniques to the larger category of orbifolds. In this paper, we exploit the connection
