A leitmotiv in geometric group theory is to make a group act on a geometric space in order to link algebraic properties of with geometric properties of . 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 and turns out to be stronger when is non-positively curved in some sense. A possible definition of geodesic space of curvature bounded above is given by inequality:
Definition: Let be a geodesic space and . For convenience, let be the only (up to isometry) simply connected Riemannian surface of constant curvature and let denote its diameter; therefore, is just the hyperbolic plane , the euclidean plane or the sphere with a rescaling metric and is finite only if .
For any geodesic triangle – that is a union of three geodesics , and – of diameter at most , there exists a comparison triangle in (unique up to isometry) such that
, and .
We say that satisfies inequality if for every
We say that is a space if every geodesic triangle of diameter at most satisfies inequality, and that is of curvature at most if it is locally .
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 if and only if its sectional curvature is bounded above by . 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 acts geometrically on a metric space , then is finitely-generated and there exists a quasi-isometry between and . See [BH, proposition I.8.19].
Let us say that a group is if it acts geometrically on some space. Notice that, if is a space, then is a space. Therefore, a group is either or or .
According to the remark made at the beginning of this note, and properties should give more information on our group than property. In fact, we are able to prove the following result:
Theorem: Any finitely-presented group is .
Sketch of proof. Let be a finitely-presented group. If is a Cayley complex of , it is know that the natural action is geometric – see Lyndon and Schupp’s book, Combinatorial group theory, section III.4. Now, the barycentric subdivision of is a flag complex of dimension two. According to Berestovskii’s theorem mentionned in [BH, theorem II.5.18], the right-angled spherical complex associated to is a (complete) space. Of course, the action is again geometric, since the underlying CW complex is just .
Another important way to link a group with a geometric space is to write as a fundamental group. As above, we can prove the following result:
Theorem: Any group is the fundamental group of a space . Moreover, if is finitely-presented, can be supposed compact.
Sketch of proof. Let be the CW complex associated to a presentation of ; then and 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 associated to the barycentric subdivision of is a space, and its fundamental group is again isomorphic to , since the underlying CW complex is just .
On the other hand, many properties are known for 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 groups are Gromov-hyperbolic. However, it is an open question to know whether or not hyperbolic groups are , or even .
Let us conclude this note by noticing that, for a fixed space , 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 , and are the only groups acting freely by homeomorphisms on the -dimensional sphere .