In mathematics, a CAT(0) group is a finitely generated group with a group action on a CAT(0) space that is geometrically proper, cocompact, and isometric. They form a possible notion of non-positively curved group in geometric group theory.
Definition Let be a group. Then is said to be a CAT(0) group if there exists a metric space and an action of on such that:
is a CAT(0) metric space The action of on is by isometries, i.e. it is a group homomorphism The action of on is geometrically proper (see below) The action is cocompact: there exists a compact subset whose translates under together cover , i.e.
An group action on a metric space satisfying conditions 2 - 4 is sometimes called geometric.
This definition is analogous to one of the many possible definitions of a Gromov-hyperbolic group, where the condition that is CAT(0) is replaced with Gromov-hyperbolicity of . However, contrarily to hyperbolicity, CAT(0)-ness of a space is not a quasi-isometry invariant, which makes the theory of CAT(0) groups a lot harder.
CAT(0) space
Metric properness The suitable notion of properness for actions by isometries on metric spaces differs slightly from that of a properly discontinuous action in topology. An isometric action of a group on a metric space is said to be geometrically proper if, for every x\in X, there exists such that is finite.
Since a compact subset of can be covered by finitely many balls such that has the above property, metric properness implies proper discontinuity. However, metric properness is a stronger condition in general. The two notions coincide for proper metric spaces.
If a group acts (geometrically) properly and cocompactly by isometries on a length space , then is actually a proper geodesic space (see metric Hopf-Rinow theorem), and is finitely generated (see Švarc-Milnor lemma). In particular, CAT(0) groups are finitely generated, and the space involved in the definition is actually proper.
Examples
CAT(0) groups
Finite groups are trivially CAT(0), and finitely generated abelian groups are CAT(0) by acting on euclidean spaces. Crystallographic groups Fundamental groups of compact Riemannian manifolds having non-positive sectional curvature are CAT(0) thanks to their action on the universal cover, which is a Cartan-Hadamard manifold. More generally, fundamental groups of compact, locally CAT(0) metric spaces are CAT(0) groups, as a consequence of the metric Cartan-Hadamard theorem. This includes groups whose Dehn complex can wear a piecewise-euclidean metric of non-positive curvature. Examples of these are provided by presentations satisfying small cancellation conditions. Any finitely presented group is a quotient of a CAT(0) group (in fact, of a fundamental group of a 2-dimensional CAT(-1) complex) with finitely generated kernel. Free products of CAT(0) groups and free amalgamated products of CAT(0) groups over finite or infinite cyclic subgroups are CAT(0). Coxeter groups are CAT(0), and act properly cocompactly on CAT(0) cube complexes. Fundamental groups of hyperbolic knot complements. , the automorphism group of the free group of rank 2, is CAT(0). The braid groups , for , are known to be CAT(0). It is conjectured that all braid groups are CAT(0). Limit groups over free groups are CAT(0) with isolated flats.
Non-CAT(0) groups
Mapping class groups of closed surfaces with genus , or surfaces with genus and nonempty boundary or at least two punctures, are not CAT(0). Some free-by-cyclic groups cannot act properly by isometries on a CAT(0) space, although they have quadratic isoperimetric inequality. Automorphism groups of free groups of rank have exponential Dehn function, and hence (see below) are not CAT(0).
Properties
Properties of the group Let be a CAT(0) group. Then:
There are finitely many conjugacy classes of finite subgroups in . In particular, there is a bound for cardinals of finite subgroups of . The solvable subgroup theorem: any solvable subgroup of is finitely generated and virtually free abelian. Moreover, there is a finite bound on the rank of free abelian subgroups of . If is infinite, then contains an element of infinite order. If is a free abelian subgroup of and is a finitely generated subgroup of containing in its center, then a finite index subgroup of splits as a direct product . The Dehn function of is at most quadratic. has a finite presentation with solvable word problem and conjugacy problem.
Properties of the action
Let be a group acting properly cocompactly by isometries on a CAT(0) space .
Any finite subgroup of fixes a nonempty closed convex set. For any infinite order element , the set of elements such that is minimal is a nonempty, closed, convex, -invariant subset of , called the minimal set of . Moreover, it splits isometrically as a (l²) direct product of a closed convex set and a geodesic line, in such a way that acts trivially on the factor and by translation on the factor. A geodesic line on which acts by translation is always of the form , , and is called an axis of . Such an element is called hyperbolic. The flat torus theorem: any free abelian subgroup leaves invariant a subspace isometric to , and acts cocompactly on (hence the quotient is a flat torus). In certain situations, a splitting of as a cartesian product induces a splitting of the space and of the action.
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