We study convergence group actions on continua, and give a topological criterion which ensures that every global cut point is a parabolic fixed point. As a corollary, we deduce that if G is a relatively hyperbolic group whose boundary is connected, and such that each peripheral subgroup is finitely presented, one-or-two ended and contains no infinite torsion subgroup, then every global cut point of the boundary of G is a parabolic fixed point. We discuss how this relates to other connectedness properties of boundaries.
1991 Mathematics Subject Classification : 20F32.
Trans. Amer. Math. Soc. 351 (1999) 3673-3686.