Asymptotically CAT(0) Groups

Details Asymptotically CAT(0) Groups Asymptotically CAT(0) Groups

2008-10-22

arxiv.org/abs/0810.4046v2

Mathematics

Group Theory

This is the version for publication. I have added some open questions in the final paragragh.

Scientific paper

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.

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