Relatively hyperbolic groups with semistable peripheral subgroups

Abstract

Suppose G is a finitely presented group that is hyperbolic relative to [Formula: see text] a finite collection of finitely generated proper subgroups of G. Our main theorem states that if each [Formula: see text] has semistable fundamental group at [Formula: see text], then G has semistable fundamental group at [Formula: see text]. The problem reduces to the case when G and the members of [Formula: see text] are all one ended and finitely presented. In that case, if the boundary [Formula: see text] has no cut point, then G was already known to have semistable fundamental group at [Formula: see text]. We consider the more general situation when [Formula: see text] contains cut points.