Peripheral Structures of Relatively Hyperbolic Groups: a Geometric Approach from Boundaries

In the 20 years since their introduction by Gromov, relatively hyperbolic groups have been receiving a great deal of attention. This class of groups encompasses many naturally occurring groups, such as geometrically finite Kleinian groups, word hyperbolic groups, limit groups, CAT(0)-groups with isolated flats and many others. The objective of this book focuses on the study of relatively hyperbolic groups and their quasiconvex subgroups. We introduce and characterize a class of parabolically extended structures for relatively hyperbolic groups. A characterization of relative quasiconvexity with respect to parabolically extended structures is obtained. The connection between the existence of a minimal relatively hyperbolic structure on a given group and the action on its Floyd boundary is examined. A result of independent interest in the thesis is that a separable subgroup has the bounded packing property. This implies that the property is true for each subgroup of a polycyclic group, answering a question of Hruska and Wise.