We study the general theory of asymptotically CAT(0) groups, explaining why such a group has finitely many conjugacy classes of finite subgroups, is $F_\infty$ and has solvable word problem. We provide techniques to combine asymptotically CAT(0) groups via direct products, amalgams and HNN extension...
We study the general theory of asymptotically CAT(0) groups, explaining why such a group has finitely many conjugacy classes of finite subgroups, is $F_\infty$ 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 $PSL(2,\mathbb{R})$ is shown to be an asymptotically CAT(0) metric space. Therefore, co-compact lattices in $\widetilde{PSL(2,\mathbb{R})}$ provide the first examples of asymptotically CAT(0) groups which are neither CAT(0) nor hyperbolic. Another source of examples is shown to be the class of relatively hyperbolic groups.