Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
ON VOLUMES OF HYPERBOLIC ORBIFOLDS ILESANMI ADEBOYE AND GUOFANG WEI
 

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
n-orbifold 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 n-orbifold Q is the quotient Hn/, where Hn denotes hyperbolic n-space
and represents a discrete group of orientation-preserving isometries of Hn. When con-
tains no elements of finite order, Q is a hyperbolic n-manifold. The problem of determining
the hyperbolic n-orbifold and n-manifold 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 3-manifolds [4] and 3-orbifolds [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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics