This post is partially based on portions of Martin Bridson’s 2006 ICM article.
In the last post we explained what it means to say a space is CAT(0) or CAT(-1). Now we turn to groups.
CAT(0) and CAT(-1) groups
A group is CAT(0) or CAT(-1) when it acts properly and cocompactly by isometries on a CAT(0) or CAT(-1) space, respectively.
An action of a group by isometries on a metric space is proper when about every point in , there is an open ball that is translated fully off itself by all but finitely many group elements. To avoid confusion, and because they are also important, let us review some related conditions. As explained by Bridson and Haefliger (page 132) when is proper (i.e. every closed ball is compact), this is equivalent to requiring that for every compact subset of , the translates of by all but finitely many groups are disjoint from . If about every point in , there is an open ball that is translated fully off itself by all group elements apart from the identity, then would be acting properly discontinuously and, assuming is connected and locally path connected Hausdorff space (which a CAT(0) space always is), would be a covering map.
An action of a group on a space is cocompact when the quotient is compact.
Examples of CAT(-1) groups include —
- Finite groups
- Free groups of finite rank (or, more generally, finitely generated groups quasi-isometric to a free group — see Theorem 37 in these notes on lectures by Whyte)
- Fundamental groups of closed surfaces of genus at least 2
- Hyperbolic or finite triangle groups (such as the (2,3,7) example pictured below)
Examples of CAT(0) groups include —
- All CAT(-1) groups
- Euclidean triangle groups
- Right–angled Artin groups
- Coxeter groups (this was Moussong’s PhD thesis; there is also an account in Davis’ book)
- Small–cancellation groups — N. Brady and McCammond, (to appear) and Wise — see Section 9 of this survey by McCammond
- Limit groups — Alibegovic and Bestvina
- Aut() and Aut() — Piggott, Ruane and Walsh (in contrast to Aut() for — Gersten)
Open question. Bridson asks which free–by–cyclic groups are CAT(0)? There are examples that are and others that are not.
The conditions of being CAT(0) or CAT(-1) for a group are not entirely satisfying. For one, they are not intrinsic properties of a group — they make reference to a space on which it acts. And, perhaps worse, the CAT(0) condition is not a quasi–isometry invariant for groups. [We presume quasi-isometry invariance of the CAT(-1) condition for groups is an open question, given that it is unknown whether hyperbolic groups are CAT(-1).] Examples of two quasi-isometric groups, only one of which is CAT(0), can be may be constructed using the fundamental groups of graph manifolds (M. Kapovich and Leeb) or of Seifert fibre spaces (page 258 of Bridson and Haefliger). [Thanks, PRW for the references.]
Driven by such concerns, efforts have been made to isolate essential features (such as those identified in our previous post) of non-positive curvature in metric spaces and recognise what they mean for groups which act on those spaces. We will take up this theme in our next post.