M.Sc Thesis

M.Sc StudentElblinger Yair
SubjectHNN Extensions of CAT(0) Groups over Free Abelian Groups
DepartmentDepartment of Mathematics
Supervisor PROF. Michah Sageev


In this work we study HNN-extensions over  of   CAT(0)  torsion free groups and general HNN-extensions of finitely generated free abelian groups.

We answer the question: When is the group  CAT(0)?  On those cases when is CAT(0)  we specifically construct a  CAT(0) space on which it acts and describe the appropriate action. The fact that we have HNN-extensions rather then free product with amalgamation makes this question more delicate and leads us to careful geometric constructions.

We study the natural inclusion  where  is a CAT(0) group. We prove that some different conditions ensure us that the inclusion is a quasi-isometric embedding, more precisely, a quasi-convex embedding with respect to the generating set   where generate   and  is the conjugating element. This is done by studying the asymptotic behavior of the axes of the elements we conjugate (in the appropriate CAT(0) space). Here as before we construct a metric space (not necessarily CAT(0)) on which  acts. Some lemmas are proven on the way, which are in themselves interesting, in particular we prove a theorem about the algebraic relation between two elements in a CAT(0) group that have non-divergent axes.

We answer the question: What is the influence of an HNN-extension over  on the number of ends of a finitely generated group ? The answer depends on the number of ends of  (it is not determinate when  has infinite number of ends).