CAT(0) group

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:

1. is a CAT(0) metric space
2. The action of on is by isometries, i.e. it is a group homomorphism
3. The action of on is geometrically proper (see below)
4. 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 <chem>x\in X</chem>, 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 Å varc-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.

References