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 $\mathbb{S}^n$.