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

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

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 ${\displaystyle X}$ is CAT(0) is replaced with Gromov-hyperbolicity of ${\displaystyle X}$. 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.^[1] An isometric action of a group ${\displaystyle G}$ on a metric space ${\displaystyle X}$ is said to be geometrically proper if, for every ${\displaystyle {\ce {x\in X}}}$, there exists ${\displaystyle r>0}$ such that ${\displaystyle \{g\in G|B(x,r)\cap g\cdot B(x,r)\neq \emptyset \}}$is finite.

Since a compact subset ${\displaystyle K}$ of ${\displaystyle X}$ can be covered by finitely many balls ${\displaystyle B(x_{i},r_{i})}$ such that ${\displaystyle B(x_{i},2r_{i})}$ 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 ${\displaystyle G}$ acts (geometrically) properly and cocompactly by isometries on a length space ${\displaystyle X}$, then ${\displaystyle X}$ is actually a proper geodesic space (see metric Hopf-Rinow theorem), and ${\displaystyle G}$ is finitely generated (see Švarc-Milnor lemma). In particular, CAT(0) groups are finitely generated, and the space ${\displaystyle X}$ 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.^[2]
• 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.^[2]
• Free products of CAT(0) groups and free amalgamated products of CAT(0) groups over finite or infinite cyclic subgroups are CAT(0).^[3]
• Coxeter groups are CAT(0), and act properly cocompactly on CAT(0) cube complexes.^[4]
• Fundamental groups of hyperbolic knot complements.^[2]
• ${\displaystyle \mathrm {Aut} (F_{2})}$, the automorphism group of the free group of rank 2, is CAT(0).^[5]
• The braid groups ${\displaystyle B_{n}}$, for ${\displaystyle n\leq 6}$, are known to be CAT(0). It is conjectured that all braid groups are CAT(0).^[6]
• Limit groups over free groups are CAT(0) with isolated flats.^[7]

Non-CAT(0) groups

• Mapping class groups of closed surfaces with genus ${\displaystyle \geq 3}$, or surfaces with genus ${\displaystyle \geq 2}$ and nonempty boundary or at least two punctures, are not CAT(0).^[8]
• Some free-by-cyclic groups cannot act properly by isometries on a CAT(0) space,^[9] although they have quadratic isoperimetric inequality.^[10]
• Automorphism groups of free groups of rank ${\displaystyle \geq 3}$ have exponential Dehn function, and hence (see below) are not CAT(0).^[11]

Properties

Properties of the group

Let ${\displaystyle G}$ be a CAT(0) group. Then:

• There are finitely many conjugacy classes of finite subgroups in ${\displaystyle G}$.^[12] In particular, there is a bound for cardinals of finite subgroups of ${\displaystyle G}$.
• The solvable subgroup theorem: any solvable subgroup of ${\displaystyle G}$ is finitely generated and virtually free abelian. Moreover, there is a finite bound on the rank of free abelian subgroups of ${\displaystyle G}$.^[8]
• If ${\displaystyle G}$ is infinite, then ${\displaystyle G}$ contains an element of infinite order.^[13]
• If ${\displaystyle A}$ is a free abelian subgroup of ${\displaystyle G}$ and ${\displaystyle C}$ is a finitely generated subgroup of ${\displaystyle G}$ containing ${\displaystyle A}$ in its center, then a finite index subgroup ${\displaystyle D}$ of ${\displaystyle C}$ splits as a direct product ${\displaystyle D\cong A\times B}$.^[14]
• The Dehn function of ${\displaystyle G}$ is at most quadratic.^[15]
• ${\displaystyle G}$ has a finite presentation with solvable word problem and conjugacy problem.^[15]

Properties of the action

Let ${\displaystyle G}$ be a group acting properly cocompactly by isometries on a CAT(0) space ${\displaystyle X}$.

• Any finite subgroup of ${\displaystyle G}$ fixes a nonempty closed convex set.
• For any infinite order element ${\displaystyle g\in G}$, the set ${\displaystyle \min(g)}$ of elements ${\displaystyle x\in X}$ such that ${\displaystyle d(g\cdot x,x)>0}$ is minimal is a nonempty, closed, convex, ${\displaystyle g}$-invariant subset of ${\displaystyle X}$, called the minimal set of ${\displaystyle g}$. Moreover, it splits isometrically as a (l²) direct product ${\displaystyle \min(g)=A\times \mathbb {R} }$ of a closed convex set ${\displaystyle A\subset X}$ and a geodesic line, in such a way that ${\displaystyle g}$ acts trivially on the ${\displaystyle A}$ factor and by translation on the ${\displaystyle \mathbb {R} }$ factor. A geodesic line on which ${\displaystyle g}$ acts by translation is always of the form ${\displaystyle \{a\}\times \mathbb {R} }$, ${\displaystyle a\in A}$, and is called an axis of ${\displaystyle g}$. Such an element is called hyperbolic.
• The flat torus theorem: any free abelian subgroup ${\displaystyle \mathbb {Z} ^{n}\cong A\subset G}$ leaves invariant a subspace ${\displaystyle F\subset X}$ isometric to ${\displaystyle \mathbb {R} ^{n}}$, and ${\displaystyle A}$ acts cocompactly on ${\displaystyle F}$ (hence the quotient ${\displaystyle F/A}$ is a flat torus).^[8]
• In certain situations, a splitting of ${\displaystyle G\cong G_{1}\times G_{2}}$ as a cartesian product induces a splitting of the space ${\displaystyle X\cong X_{1}\times X_{2}}$ and of the action.^[14]