We prove a true bootstrapping result for convergence groups acting on a Peano continuum. We give an example of a Kleinian group which is the amalgamation of two closed hyperbolic surface groups along a simple closed curve. The limit set is the closure of a “tree of circles" (adjacent circles meeting in pairs of points). We alter the action of on its limit set such that 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.
"Bootstrapping in convergence groups." Algebr. Geom. Topol. 5 (2) 751 - 768, 2005. https://doi.org/10.2140/agt.2005.5.751