Let 
denote the exponent of convergence of the Poincare series
where z,w are fixed points of 
and (a,b)
is the hyperbolic distance from a to b.
Let 
denote the Hausdorff dimension of the limit set 
.
By work of Patterson
[11],
[12]
and Sullivan
[17],
[18],
and
as long as 
.
Thus our universal lower bound for 
implies
a universal upper bound 
for
and .