For $p, q ge 2$ and max${p, q} ge 3$ we denote by $c(p, q)$ the smallest number with the following property. If $f$ and $g$ are elliptic M"{o}bius transformations of orders $p$ and $q$ and if the hyperbolic distance $delta(f, g)$ between their axes is at least $c(p, q)$, then the group $langle f, grangle$ is discrete, nonelementary andisomorphic to the free product $Z_p * Z_q$. We prove here that [ cos h (c(p, q)) = {cos(pi/p) cos (pi/q)+1 over sin (pi/p)sin (pi/q)}. ] This value is attained in the $(p, q, infty)4-triangle group. We give an application concerning the commutator parameter of the free product of cyclic groups. |

__Keywords__

Rungek-Kutta, explicit, coefficients, dual precision

__Math Review Classification__

Primary 30F40, 20H10

__Last Updated__

2/4/97

__Length__

7 pages

__Availability__

This article is available in: