CAT(0) group explained

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

such that:

is a CAT(0) metric space

The action of

is by isometries, i.e. it is a group homomorphism

G\longrightarrowIsom(X)

The action of

is geometrically proper (see below)

The action is cocompact: there exists a compact subset

K\subsetX

whose translates under

together cover

, i.e.

X=G ⋅ K=cup_g\ing ⋅ 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

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

See main article: article and 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

on a metric space

is said to be geometrically proper if, for every x\in X, there exists

r>0

such that

\{g\inG|B(x,r)\capg ⋅ B(x,r) ≠ \emptyset\}

is finite.

Since a compact subset

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

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.^[1]
Fundamental groups of hyperbolic knot complements.

Aut(F₂₎

, the automorphism group of the free group of rank 2, is CAT(0).^[2]

B_n

, for

n\le6

, are known to be CAT(0). It is conjectured that all braid groups are CAT(0).^[3]

Non-CAT(0) groups

Mapping class groups of closed surfaces with genus

\ge3

, or surfaces with genus

\ge2

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,^[4] although they have quadratic isoperimetric inequality.^[5]
Automorphism groups of free groups of rank

\ge3

have exponential Dehn function, and hence (see below) are not CAT(0).^[6]

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

is infinite, then

contains an element of infinite order.^[7]

is a free abelian subgroup of

and

is a finitely generated subgroup of

containing

in its center, then a finite index subgroup

splits as a direct product

D\congA x B

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

g\inG

, the set

min(g)

of elements

x\inX

such that

d(g ⋅ x,x)>0

is minimal is a nonempty, closed, convex,

-invariant subset of

, called the minimal set of

. Moreover, it splits isometrically as a (l²) direct product

min(g)=A x \R

of a closed convex set

A\subsetX

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

\{a\} x \R

a\inA

, and is called an axis of

. Such an element is called hyperbolic.

The flat torus theorem: any free abelian subgroup

\Zⁿ\congA\subsetG

leaves invariant a subspace

F\subsetX

isometric to

\Rⁿ

, and

acts cocompactly on

(hence the quotient

F/A

is a flat torus).

In certain situations, a splitting of

G\congG_{1 x}G₂

as a cartesian product induces a splitting of the space

X\congX_{1 x}X₂

and of the action.

References

Niblo . G. A. . Reeves . L. D. . 2003-01-27 . Coxeter Groups act on CAT(0) cube complexes . Journal of Group Theory . 6 . 3 . 10.1515/jgth.2003.028 . 1433-5883.
Piggott . Adam . Ruane . Kim . Walsh . Genevieve . 2010 . The automorphism group of the free group of rank 2 is a CAT(0) group . Michigan Mathematical Journal . 59 . 2 . 297–302 . 10.1307/mmj/1281531457 . 0026-2285. 0809.2034 .
Haettel . Thomas . Kielak . Dawid . Schwer . Petra . 2016-06-01 . The 6-strand braid group is CAT(0) . Geometriae Dedicata . en . 182 . 1 . 263–286 . 10.1007/s10711-015-0138-9 . 1572-9168.
Gersten . S. M. . 1994 . The Automorphism Group of a Free Group Is Not a $\operatorname(0)$ Group . Proceedings of the American Mathematical Society . 121 . 4 . 999–1002 . 10.2307/2161207 . 0002-9939.
Web site: Bridson . Martin . Groves . Daniel . 2010 . The quadratic isoperimetric inequality for mapping tori of free group automorphisms . 2024-11-19 . American Mathematical Society . en.
Hatcher . Allen . Vogtmann . Karen . 1996-04-01 . Isoperimetric inequalities for automorphism groups of free groups . Pacific Journal of Mathematics . 173 . 2 . 425–441 . 0030-8730.
Swenson . Eric L. . 1999 . A cut point theorem for $\rm(0)$ groups . Journal of Differential Geometry . 53 . 2 . 327–358 . 10.4310/jdg/1214425538 . 0022-040X.