A leitmotiv in geometric group theory is to make a group G act on a geometric space X in order to link algebraic properties of G with geometric properties of X. Here, the expression geometric space is intentionally vague: it goes from rather combinatorial objects, like simplicial trees, to rather geometric objects, like Riemannian manifolds. However, it is worth noticing that the link between G and X turns out to be stronger when X is non-positively curved in some sense. A possible definition of geodesic space of curvature bounded above is given by CAT(\kappa) inequality:

Definition: Let X be a geodesic space and \kappa \in \mathbb{R}. For convenience, let M_{\kappa} be the only (up to isometry) simply connected Riemannian surface of constant curvature \kappa and let D_{\kappa} denote its diameter; therefore, M_{\kappa} is just the hyperbolic plane M_{-1}, the euclidean plane M_0 or the sphere M_1 with a rescaling metric and D_{\kappa} is finite only if \kappa >0.

For any geodesic triangle \Delta=\Delta(x,y,z) – that is a union of three geodesics [x,y], [y,z] and [x,z] – of diameter at most 2 D_{\kappa}, there exists a comparison triangle \overline{\Delta}= \overline{\Delta}(\overline{x},\overline{y},\overline{z}) in M_{\kappa} (unique up to isometry) such that

d(\overline{x}, \overline{y})= d(x,y),  d(\overline{y}, \overline{z})= d(y,z)  and  d(\overline{x}, \overline{z})=d(x,z).

We say that \Delta satisfies CAT(\kappa) inequality if for every a,b \in \Delta

d(a,b) \leq d( \overline{a}, \overline{b}).

We say that X is a CAT(\kappa) space if every geodesic triangle of diameter at most 2 D_{\kappa} satisfies CAT(\kappa) inequality, and that X is of curvature at most \kappa if it is locally CAT(\kappa).

This definition of curvature bounded above is good enough to agree with sectional curvature of Riemannian manifolds: A Riemannian manifold is of curvature at most \kappa if and only if its sectional curvature is bounded above by \kappa. For more information, see Bridson and Haefliger’s book, Metric spaces of nonpositive curvature, theorem I.1.A6.

From now on, let us consider a nice kind of action on geodesic spaces:

Definition: A group acts geometrically on a metric space whenever the action is properly discontinuous and cocompact.

Such actions are fundamental in geometric group theory, notably because of Milnor-Svarc theorem: if a group G acts geometrically on a metric space X, then G is finitely-generated and there exists a quasi-isometry between G and X. See [BH, proposition I.8.19].

Let us say that a group is CAT(\kappa) if it acts geometrically on some CAT(\kappa) space. Notice that, if (X,d) is a CAT(\kappa) space, then (X, \sqrt{ | \kappa | } \cdot d ) is a CAT( \kappa / |\kappa|) space. Therefore, a CAT(\kappa) group is either CAT(-1) or CAT(0) or CAT(1).

According to the remark made at the beginning of this note, CAT(-1) and CAT(0) properties should give more information on our group than CAT(1) property. In fact, we are able to prove the following result:

Theorem: Any finitely-presented group is CAT(1).

Sketch of proof. Let G be a finitely-presented group. If X is a Cayley complex of G, it is know that the natural action G \curvearrowright X is geometric – see Lyndon and Schupp’s book, Combinatorial group theory, section III.4. Now, the barycentric subdivision X' of X is a flag complex of dimension two. According to Berestovskii’s theorem mentionned in [BH, theorem II.5.18], the right-angled spherical complex Y associated to X' is a (complete) CAT(1) space. Of course, the action G \curvearrowright Y is again geometric, since the underlying CW complex is just X'. \square

Another important way to link a group G with a geometric space is to write G as a fundamental group. As above, we can prove the following result:

Theorem: Any group G is the fundamental group of a CAT(1) space X. Moreover, if G is finitely-presented, X can be supposed compact.

Sketch of proof. Let X be the CW complex associated to a presentation of G; then G \simeq \pi_1(X) and X is compact if the presentation were finite – see Lyndon and Schupp’s book, Combinatorial group theory, section III.4. As above, the right-angled spherical complex Y associated to the barycentric subdivision X' of X is a CAT(1) space, and its fundamental group is again isomorphic to G, since the underlying CW complex is just X'. \square

On the other hand, many properties are known for CAT(0) groups; the usual reference on the subjet is Bridson and Haefliger’s book, Metric spaces of nonpositive curvature. Furthermore, is not difficult to prove that CAT(-1) groups are Gromov-hyperbolic. However, it is an open question to know whether or not hyperbolic groups are CAT(-1), or even CAT(0).

Let us conclude this note by noticing that, for a fixed CAT(1) space X, it is possible that only few groups are able to act on it. For example, we proved in our previous note Brouwer’s Topological Degree (IV): Jordan Curve Theorem that, for any even number n, \{1 \} and \mathbb{Z}_2 are the only groups acting freely by homeomorphisms on the n-dimensional sphere \mathbb{S}^n.

 

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