  We prove a true bootstrapping result for convergence groups acting on a Peano
continuum. We give an example of a Kleinian group H which is the amalgamation
of two closed hyperbolic surface groups along a simple closed curve. The limit
set Lambda H is the closure of a `tree of circles' (adjacent circles meeting in
pairs of points). We alter the action of H on its limit set such that H no
longer acts as a convergence group, but the stabilizers of the circles remain
unchanged, as does the action of a circle stabilizer on said circle. This is
done by first separating the circles and then gluing them together backwards.
