Asymptotically CAT(0) Groups

TL;DR

This paper develops the theory of asymptotically CAT(0) groups, showing their algebraic and geometric properties, providing new examples, and techniques for constructing such groups from known classes.

Contribution

It introduces the concept of asymptotically CAT(0) groups, proves their key properties, and constructs new examples including certain lattices and relatively hyperbolic groups.

Findings

Asymptotically CAT(0) groups have finitely many conjugacy classes of finite subgroups.

They are of type $F_$ and have solvable word problem.

Examples include lattices in the universal cover of $PSL(2,R)$ and relatively hyperbolic groups.

Abstract

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.