In this dissertation, we introduce the class of asymptotically CAT(0) groups, the principal objective being to study their algebraic properties and provide examples. The initial focus is on δ-CAT(0) groups, which form a special class of asymptotically CAT(0) groups. These have many desirable algebraic properties, in particular, they are semihyperbolic and satisfy Novikov’s Conjecture on Higher Signatures. We observe that there are examples of metric spaces which are asymptotically CAT(0) but not δ-CAT(0).
We proceed to study the general theory of asymptotically CAT(0) groups, explaining why such a group has finitely many conjugacy classes of finite subgroups, is F∞ and has solvable word problem. We provide techniques to combine asymptotically CAT(0) groups via direct products, amalgams and HNN extensions.
The universal cover of the Lie group PSL(2,ℝ) is shown to be an asymptotically CAT(0) metric space. Therefore, cocompact lattices in this universal cover provide the first examples of asymptotically CAT(0) groups which are neither CAT(0) nor hyperbolic. Another potential rich source of examples is the class of relatively hyperbolic groups.
We conclude with a selection of interesting questions which arise out of this dissertation.