On the Discreteness of the Free Product of Finite Cyclic groups

F.W. Gehring, C. Maclachlan, G.J. Martin


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.

Rungek-Kutta, explicit, coefficients, dual precision

Math Review Classification
Primary 30F40, 20H10

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7 pages

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