Boundaries of groups

• Other Authors: Ruane, Kim E.
• Format: Others
• Language: English
• Subjects: Mathematics
• Online Access: http://purl.flvc.org/fsu/lib/digcoll/etd/3088922

In recent years, the theory of infinite groups has been revolutionized by the introduction of geometric methods. In his foundational paper, "Hyperbolic Groups", Gromov outlines a geometric group theory which provides tools for studying a wide class of groups meant to generalize the classical groups coming from Riemannian geometry. In this setting, the metric geometry of the space is used to study the algebraic properties of the group. One aspect of the metric geometry is the behavior of geodesic rays in the space. A technique used for studying this behavior is to compactify the space by adding the endpoints of geodesic rays--i.e. the boundary of the space. === Several new theorems in group theory were proven only after the introduction of these geometric methods--for instance, the Scott conjecture--and many known theorems can be given new, elegant geometric proofs. With the success of this approach, Gromov wrote a second paper which gives certain minimum requirements for a theory including certain non-positively curved groups. === The first task is to define a notion of non-positive curvature that will generalize the classical Riemannian notion. One proposed notion goes back to the work of Alexandroff and Topogonov wherein they compare the triangles in a given geometry to the triangles in Euclidean geometry and ask that those in the former be as least as thin as those in the latter. Then a class of non-positively curved groups can be defined as those that act geometrically on one of these non-positively curved spaces. === My research has focused on studying the boundary of the non-positively curved spaces which admit geometric actions by a group. The overriding question is a question in Gromov's second paper: If a group acts geometrically on two such spaces, then do they have homeomorphic boundaries? === Source: Dissertation Abstracts International, Volume: 57-04, Section: B, page: 2612. === Major Professor: Philip Bowers. === Thesis (Ph.D.)--The Florida State University, 1996.