CAT(0) group

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 $G\; \backslash longrightarrow\; \backslash mathrm\{Isom\}(X)$
3. The action of $G$ on $X$ is geometrically proper (see below)
4. The action is cocompact: there exists a compact subset $K\backslash subset\; X$ whose translates under $G$ together cover $X$, i.e. $X\; =\; G\backslash cdot\; K\; =\; \backslash bigcup\_\{g\backslash in\; G\}\; g\backslash cdot\; K$

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.

{{Main article|CAT(k) space}}

The suitable notion of properness for actions by isometries on metric spaces differs slightly from that of a properly discontinuous action in topology.{{Citation |last1=Bridson |first1=Martin R. |title=Group Actions and Quasi-Isometries |date=1999 |work=Metric Spaces of Non-Positive Curvature |pages=131–156 |editor-last=Bridson |editor-first=Martin R. |url=https://link.springer.com/chapter/10.1007/978-3-662-12494-9_8 |access-date=2024-11-19 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-12494-9_8 |isbn=978-3-662-12494-9 |last2=Haefliger |first2=André |editor2-last=Haefliger |editor2-first=André}} 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 $\backslash \{g\backslash in\; G\; |\; B(x,\; r)\backslash cap\; g\backslash cdot\; B(x,\; r)\; \backslash neq\; \backslash emptyset\backslash \}$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.

• 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.{{Citation |last1=Bridson |first1=Martin R. |title=Mк-Polyhedral Complexes of Bounded Curvature |date=1999 |work=Metric Spaces of Non-Positive Curvature |pages=205–227 |editor-last=Bridson |editor-first=Martin R. |url=https://link.springer.com/chapter/10.1007/978-3-662-12494-9_13 |access-date=2024-11-19 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-12494-9_13 |isbn=978-3-662-12494-9 |last2=Haefliger |first2=André |editor2-last=Haefliger |editor2-first=André}}
• 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).{{Citation |last1=Bridson |first1=Martin R. |title=Gluing Constructions |date=1999 |work=Metric Spaces of Non-Positive Curvature |pages=347–366 |editor-last=Bridson |editor-first=Martin R. |url=https://link.springer.com/chapter/10.1007/978-3-662-12494-9_19 |access-date=2024-11-19 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-12494-9_19 |isbn=978-3-662-12494-9 |last2=Haefliger |first2=André |editor2-last=Haefliger |editor2-first=André}}
• Coxeter groups are CAT(0), and act properly cocompactly on CAT(0) cube complexes.{{Cite journal |last1=Niblo |first1=G. A. |last2=Reeves |first2=L. D. |date=2003-01-27 |title=Coxeter Groups act on CAT(0) cube complexes |journal=Journal of Group Theory |volume=6 |issue=3 |doi=10.1515/jgth.2003.028 |s2cid=17040423 |issn=1433-5883}}
• Fundamental groups of hyperbolic knot complements.
• $\backslash mathrm\{Aut\}(F\_2)$, the automorphism group of the free group of rank 2, is CAT(0).{{Cite journal |last1=Piggott |first1=Adam |last2=Ruane |first2=Kim |last3=Walsh |first3=Genevieve |date=2010 |title=The automorphism group of the free group of rank 2 is a CAT(0) group |url=https://projecteuclid.org/journals/michigan-mathematical-journal/volume-59/issue-2/The-automorphism-group-of-the-free-group-of-rank-2/10.1307/mmj/1281531457.full |journal=Michigan Mathematical Journal |volume=59 |issue=2 |pages=297–302 |doi=10.1307/mmj/1281531457 |issn=0026-2285|arxiv=0809.2034 }}
• The braid groups $B\_n$, for $n\backslash le\; 6$, are known to be CAT(0). It is conjectured that all braid groups are CAT(0).{{Cite journal |last1=Haettel |first1=Thomas |last2=Kielak |first2=Dawid |last3=Schwer |first3=Petra |date=2016-06-01 |title=The 6-strand braid group is CAT(0) |url=https://link.springer.com/article/10.1007/s10711-015-0138-9 |journal=Geometriae Dedicata |language=en |volume=182 |issue=1 |pages=263–286 |doi=10.1007/s10711-015-0138-9 |issn=1572-9168|arxiv=1304.5990 }}

• Mapping class groups of closed surfaces with genus $\backslash ge\; 3$, or surfaces with genus $\backslash ge\; 2$ and nonempty boundary or at least two punctures, are not CAT(0).{{Citation |last1=Bridson |first1=Martin R. |title=The Flat Torus Theorem |date=1999 |work=Metric Spaces of Non-Positive Curvature |pages=244–259 |editor-last=Bridson |editor-first=Martin R. |url=https://link.springer.com/chapter/10.1007/978-3-662-12494-9_15 |access-date=2024-11-19 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-12494-9_15 |isbn=978-3-662-12494-9 |last2=Haefliger |first2=André |editor2-last=Haefliger |editor2-first=André}}
• Some free-by-cyclic groups cannot act properly by isometries on a CAT(0) space,{{Cite journal |last=Gersten |first=S. M. |date=1994 |title=The Automorphism Group of a Free Group Is Not a $\operatorname{Cat}(0)$ Group |url=https://www.jstor.org/stable/2161207 |journal=Proceedings of the American Mathematical Society |volume=121 |issue=4 |pages=999–1002 |doi=10.2307/2161207 |jstor=2161207 |issn=0002-9939}} although they have quadratic isoperimetric inequality.{{Cite journal |last1=Bridson |first1=Martin |last2=Groves |first2=Daniel |date=2010 |title=The quadratic isoperimetric inequality for mapping tori of free group automorphisms |url=http://www.ams.org/books/memo/0955/ |access-date=2024-11-19 |journal=Memoirs of the American Mathematical Society |volume=203 |issue=955 |doi=10.1090/S0065-9266-09-00578-X |language=en|arxiv=math/0610332 }}
• Automorphism groups of free groups of rank $\backslash ge\; 3$ have exponential Dehn function, and hence (see below) are not CAT(0).{{Cite journal |last1=Hatcher |first1=Allen |last2=Vogtmann |first2=Karen |date=1996-04-01 |title=Isoperimetric inequalities for automorphism groups of free groups |url=https://msp.org/pjm/1996/173-2/p09.xhtml |journal=Pacific Journal of Mathematics |volume=173 |issue=2 |pages=425–441 |doi=10.2140/pjm.1996.173.425 |issn=0030-8730}}

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

• There are finitely many conjugacy classes of finite subgroups in $G$.{{Citation |last1=Bridson |first1=Martin R. |title=Convexity and its Consequences |date=1999 |work=Metric Spaces of Non-Positive Curvature |pages=175–183 |editor-last=Bridson |editor-first=Martin R. |url=https://link.springer.com/chapter/10.1007/978-3-662-12494-9_10 |access-date=2024-11-19 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-12494-9_10 |isbn=978-3-662-12494-9 |last2=Haefliger |first2=André |editor2-last=Haefliger |editor2-first=André}} 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.{{Cite journal |last=Swenson |first=Eric L. |date=1999 |title=A cut point theorem for $\rm{CAT}(0)$ groups |url=https://projecteuclid.org/journals/journal-of-differential-geometry/volume-53/issue-2/A-cut-point-theorem-for-rmCAT0-groups/10.4310/jdg/1214425538.full |journal=Journal of Differential Geometry |volume=53 |issue=2 |pages=327–358 |doi=10.4310/jdg/1214425538 |issn=0022-040X}}
• 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 $D\; \backslash cong\; A\backslash times\; B$.{{Citation |last1=Bridson |first1=Martin R. |title=Isometries of CAT(0) Spaces |date=1999 |work=Metric Spaces of Non-Positive Curvature |pages=228–243 |editor-last=Bridson |editor-first=Martin R. |url=https://link.springer.com/chapter/10.1007/978-3-662-12494-9_14 |access-date=2024-11-19 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-12494-9_14 |isbn=978-3-662-12494-9 |last2=Haefliger |first2=André |editor2-last=Haefliger |editor2-first=André}}
• The Dehn function of $G$ is at most quadratic.{{Citation |last1=Bridson |first1=Martin R. |title=Non-Positive Curvature and Group Theory |date=1999 |work=Metric Spaces of Non-Positive Curvature |pages=438–518 |editor-last=Bridson |editor-first=Martin R. |url=https://link.springer.com/chapter/10.1007/978-3-662-12494-9_22 |access-date=2024-11-19 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-12494-9_22 |isbn=978-3-662-12494-9 |last2=Haefliger |first2=André |editor2-last=Haefliger |editor2-first=André}}
• $G$ has a finite presentation with solvable word problem and conjugacy problem.

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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 $g\backslash in\; G$, the set $\backslash min(g)$ of elements $x\backslash in\; X$ such that $d(g\backslash cdot\; x,\; x)\; >\; 0$ 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 $\backslash min(g)\; =\; A\backslash times\; \backslash R$ of a closed convex set $A\backslash subset\; X$ and a geodesic line, in such a way that $g$ acts trivially on the $A$ factor and by translation on the $\backslash R$ factor. A geodesic line on which $g$ acts by translation is always of the form $\backslash \{a\backslash \}\backslash times\; \backslash R$, $a\backslash in\; A$, and is called an axis of $g$. Such an element is called hyperbolic.
• The flat torus theorem: any free abelian subgroup $\backslash Z^n\; \backslash cong\; A\; \backslash subset\; G$ leaves invariant a subspace $F\backslash subset\; X$ isometric to $\backslash R^n$, and $A$ acts cocompactly on $F$ (hence the quotient $F/A$ is a flat torus).
• In certain situations, a splitting of $G\; \backslash cong\; G\_1\backslash times\; G\_2$ as a cartesian product induces a splitting of the space $X\backslash cong\; X\_1\backslash times\; X\_2$ and of the action.

Category:Geometric group theory

Category:Combinatorial group theory