CAT(0) group

Definition

Let G be a group. Then G is said to be a CAT(0) group if there exists a metric space X and an action of G on X such that:

1. X is a CAT(0) metric space
2. The action of G on X is by isometries, i.e. it is a group homomorphism G \longrightarrow \mathrm{Isom}(X)
3. The action of G on X is geometrically proper (see below)
4. The action is cocompact: there exists a compact subset K\subset X whose translates under G together cover X, i.e. X = G\cdot K = \bigcup_{g\in G} g\cdot K

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 X is CAT(0) is replaced with Gromov-hyperbolicity of X. 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

Main article: CAT(k) 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 G on a metric space X is said to be geometrically proper if, for every x\in X, there exists r 0 such that {g\in G | B(x, r)\cap g\cdot B(x, r) \neq \emptyset}is finite.

Since a compact subset K of X can be covered by finitely many balls B(x_i, r_i) such that B(x_i, 2r_i) 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 G acts (geometrically) properly and cocompactly by isometries on a length space X, then X is actually a proper geodesic space (see metric Hopf-Rinow theorem), and G is finitely generated (see Švarc-Milnor lemma). In particular, CAT(0) groups are finitely generated, and the space X 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.
• \mathrm{Aut}(F_2), the automorphism group of the free group of rank 2, is CAT(0).
• The braid groups B_n, for n\le 6, 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 \ge 3, or surfaces with genus \ge 2 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 \ge 3 have exponential Dehn function, and hence (see below) are not CAT(0).

Properties

Properties of the group

Let G be a CAT(0) group. Then:

• There are finitely many conjugacy classes of finite subgroups in G. In particular, there is a bound for cardinals of finite subgroups of G.
• The solvable subgroup theorem: any solvable subgroup of G is finitely generated and virtually free abelian. Moreover, there is a finite bound on the rank of free abelian subgroups of G.
• If G is infinite, then G contains an element of infinite order.
• If A is a free abelian subgroup of G and C is a finitely generated subgroup of G containing A in its center, then a finite index subgroup D of C splits as a direct product D \cong A\times B.
• The Dehn function of G is at most quadratic.
• G has a finite presentation with solvable word problem and conjugacy problem.

Properties of the action

Let G be a group acting properly cocompactly by isometries on a CAT(0) space X.

• Any finite subgroup of G fixes a nonempty closed convex set.
• For any infinite order element g\in G, the set \min(g) of elements x\in X such that d(g\cdot x, x) 0 is minimal is a nonempty, closed, convex, g-invariant subset of X, called the minimal set of g. Moreover, it splits isometrically as a (l²) direct product \min(g) = A\times \R of a closed convex set A\subset X and a geodesic line, in such a way that g acts trivially on the A factor and by translation on the \R factor. A geodesic line on which g acts by translation is always of the form {a}\times \R, a\in A, and is called an axis of g. Such an element is called hyperbolic.
• The flat torus theorem: any free abelian subgroup \Z^n \cong A \subset G leaves invariant a subspace F\subset X isometric to \R^n, and A acts cocompactly on F (hence the quotient F/A is a flat torus).
• In certain situations, a splitting of G \cong G_1\times G_2 as a cartesian product induces a splitting of the space X\cong X_1\times X_2 and of the action.

References

References

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6. (2016-06-01). "The 6-strand braid group is CAT(0)". Geometriae Dedicata.
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