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]

References

1. Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "Group Actions and Quasi-Isometries" , Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, pp. 131–156, doi:10.1007/978-3-662-12494-9_8, ISBN   978-3-662-12494-9 , retrieved 2024-11-19
2. Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "Mк-Polyhedral Complexes of Bounded Curvature" , Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, pp. 205–227, doi:10.1007/978-3-662-12494-9_13, ISBN   978-3-662-12494-9 , retrieved 2024-11-19
3. Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "Gluing Constructions" , Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, pp. 347–366, doi:10.1007/978-3-662-12494-9_19, ISBN   978-3-662-12494-9 , retrieved 2024-11-19
4. Niblo, G. A.; Reeves, L. D. (2003-01-27). "Coxeter Groups act on CAT(0) cube complexes". Journal of Group Theory. 6 (3). doi:10.1515/jgth.2003.028. ISSN   1433-5883. S2CID   17040423.
5. 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. arXiv: 0809.2034 . doi:10.1307/mmj/1281531457. ISSN   0026-2285.
6. Haettel, Thomas; Kielak, Dawid; Schwer, Petra (2016-06-01). "The 6-strand braid group is CAT(0)". Geometriae Dedicata. 182 (1): 263–286. arXiv: 1304.5990 . doi:10.1007/s10711-015-0138-9. ISSN   1572-9168.
7. Alibegović, Emina; Bestvina, Mladen (2006). "LIMIT GROUPS ARE CAT(0)". Journal of the London Mathematical Society. 74 (1): 259–272. doi:10.1112/S0024610706023155. ISSN   1469-7750.
8. Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "The Flat Torus Theorem" , Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, pp. 244–259, doi:10.1007/978-3-662-12494-9_15, ISBN   978-3-662-12494-9 , retrieved 2024-11-19
9. Gersten, S. M. (1994). "The Automorphism Group of a Free Group Is Not a $\operatorname{Cat}(0)$ Group" . Proceedings of the American Mathematical Society. 121 (4): 999–1002. doi:10.2307/2161207. ISSN   0002-9939. JSTOR   2161207.
10. Bridson, Martin; Groves, Daniel (2010). "The quadratic isoperimetric inequality for mapping tori of free group automorphisms". Memoirs of the American Mathematical Society. 203 (955). arXiv: math/0610332 . doi:10.1090/S0065-9266-09-00578-X . Retrieved 2024-11-19.
11. Hatcher, Allen; Vogtmann, Karen (1996-04-01). "Isoperimetric inequalities for automorphism groups of free groups". Pacific Journal of Mathematics. 173 (2): 425–441. doi:10.2140/pjm.1996.173.425. ISSN   0030-8730.
12. Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "Convexity and its Consequences" , Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, pp. 175–183, doi:10.1007/978-3-662-12494-9_10, ISBN   978-3-662-12494-9 , retrieved 2024-11-19
13. Swenson, Eric L. (1999). "A cut point theorem for $\rm{CAT}(0)$ groups". Journal of Differential Geometry. 53 (2): 327–358. doi:10.4310/jdg/1214425538. ISSN   0022-040X.
14. Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "Isometries of CAT(0) Spaces" , Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, pp. 228–243, doi:10.1007/978-3-662-12494-9_14, ISBN   978-3-662-12494-9 , retrieved 2024-11-19
15. Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "Non-Positive Curvature and Group Theory" , Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, pp. 438–518, doi:10.1007/978-3-662-12494-9_22, ISBN   978-3-662-12494-9 , retrieved 2024-11-19