Bootstrapping in convergence groups

Abstract

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.