View source for CAT(0) group

In [[mathematics]], a '''CAT(0) group''' is a [[Finitely generated group|finitely generated]] [[Group (mathematics)|group]] with a [[Group action (mathematics)|group action]] on a [[CAT(0) space]] that is geometrically proper, [[Cocompact group action|cocompact]], and [[Isometry|isometric]]. They form a possible notion of non-positively curved group in [[geometric group theory]].

== Definition ==
Let <math>G</math> be a group. Then <math>G</math> is said to be a '''CAT(0) group''' if there exists a [[metric space]] <math>X</math> and an [[Group action|action]] of <math>G</math> on <math>X</math> such that:

# <math>X</math> is a [[CAT(k) space|CAT(0) metric space]]
# The action of <math>G</math> on <math>X</math> is by [[Isometry|isometries]], i.e. it is a [[group homomorphism]] <math>G \longrightarrow \mathrm{Isom}(X)</math>
# The action of <math>G</math> on <math>X</math> is geometrically proper (see below)
# The action is '''cocompact''': there exists a [[Compact space|compact]] subset <math>K\subset X</math> whose translates under <math>G</math> together cover <math>X</math>, i.e. <math>X = G\cdot K = \bigcup_{g\in G} g\cdot K</math>

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 [[Hyperbolic group|Gromov-hyperbolic group]], where the condition that <math>X</math> is CAT(0) is replaced with [[Hyperbolic metric space|Gromov-hyperbolicity]] of <math>X</math>. 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.<ref>{{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}}</ref> An isometric action of a group <math>G</math> on a metric space <math>X</math> is said to be '''geometrically proper''' if, for every <chem>x\in X</chem>, there exists <math>r > 0</math> such that  <math>\{g\in G | B(x, r)\cap g\cdot B(x, r) \neq \emptyset\}</math>is finite.

Since a compact subset <math>K</math> of <math>X</math> can be covered by finitely many balls <math>B(x_i, r_i)</math> such that <math>B(x_i, 2r_i)</math> 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 space|proper metric spaces]].

If a group <math>G</math> acts (geometrically) properly and cocompactly by isometries on a [[length space]] <math>X</math>, then <math>X</math> is actually a proper [[Geodesic space|geodesic]] space (see [[Hopf–Rinow theorem#Variations and generalizations|metric Hopf-Rinow theorem]]), and <math>G</math> is [[Finitely generated group|finitely generated]] (see [[Švarc–Milnor lemma|Švarc-Milnor lemma]]). In particular, CAT(0) groups are finitely generated, and the space <math>X</math> involved in the definition is actually proper.

== Examples ==

=== CAT(0) groups ===

* Finite groups are trivially CAT(0), and [[Finitely generated abelian group|finitely generated abelian groups]] are CAT(0) by acting on euclidean spaces.
* [[Crystallographic group|Crystallographic groups]]
* [[Fundamental group|Fundamental groups]] of compact [[Riemannian manifold|Riemannian manifolds]] having non-positive [[sectional curvature]] are CAT(0) thanks to their action on the [[universal cover]], which is a [[Hadamard manifold|Cartan-Hadamard manifold]].
* More generally, fundamental groups of compact, locally CAT(0) metric spaces are CAT(0) groups, as a consequence of the [[Cartan–Hadamard theorem#Metric geometry|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 theory|small cancellation]] conditions.<ref name=":0">{{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}}</ref>
* Any finitely presented group is a [[Quotient group|quotient]] of a CAT(0) group (in fact, of a fundamental group of a 2-dimensional CAT(-1) complex) with finitely generated kernel.<ref name=":0" />
* [[Free product|Free products]] of CAT(0) groups and [[Free products with amalgamation|free amalgamated products]] of CAT(0) groups over finite or infinite cyclic subgroups are CAT(0).<ref>{{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}}</ref>
* [[Coxeter group|Coxeter groups]] are CAT(0), and act properly cocompactly on CAT(0) [[Cube complex|cube complexes]].<ref>{{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}}</ref>
* [[Knot group|Fundamental groups of hyperbolic knot complements]].<ref name=":0" />
* <math>\mathrm{Aut}(F_2)</math>, the [[Automorphism group of a free group|automorphism group of the free group]] of rank 2, is CAT(0).<ref>{{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 }}</ref>
* The [[Braid group|braid groups]] <math>B_n</math>, for <math>n\le 6</math>, are known to be CAT(0). It is conjectured that all braid groups are CAT(0).<ref>{{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 }}</ref>

=== Non-CAT(0) groups ===

* [[Mapping class group of a surface|Mapping class groups]] of closed surfaces with genus <math>\ge 3</math>, or surfaces with genus <math>\ge 2</math> and nonempty boundary or at least two punctures, are not CAT(0).<ref name=":1">{{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}}</ref>
* Some [[Free-by-cyclic group|free-by-cyclic]] groups cannot act properly by isometries on a CAT(0) space,<ref>{{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}}</ref> although they have quadratic isoperimetric inequality.<ref>{{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 }}</ref>
* [[Automorphism group of a free group|Automorphism groups of free groups]] of rank <math>\ge 3</math> have exponential Dehn function, and hence (see below) are not CAT(0).<ref>{{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}}</ref>

== Properties ==

=== Properties of the group ===
Let <math>G</math> be a CAT(0) group. Then:

* There are finitely many [[Conjugacy class#Conjugacy of subgroups and general subsets|conjugacy classes]] of finite subgroups in <math>G</math>.<ref>{{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}}</ref> In particular, there is a bound for [[Cardinal (mathematics)|cardinals]] of finite subgroups of <math>G</math>.
* The solvable subgroup theorem: any [[Solvable group|solvable]] subgroup of <math>G</math> is finitely generated and [[virtually]] [[Free abelian group|free abelian]]. Moreover, there is a finite bound on the [[Rank of a free module|rank]] of free abelian subgroups of <math>G</math>.<ref name=":1" />
* If <math>G</math> is infinite, then <math>G</math> contains an element of infinite order.<ref>{{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}}</ref>
* If <math>A</math> is a free abelian subgroup of <math>G</math> and <math>C</math> is a finitely generated subgroup of <math>G</math> containing <math>A</math> in its [[Center (group theory)|center]], then a finite index subgroup <math>D</math> of <math>C</math> splits as a direct product <math>D \cong A\times B</math>.<ref name=":2">{{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}}</ref>
* The [[Dehn function]] of <math>G</math> is at most quadratic.<ref name=":3">{{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}}</ref>
* <math>G</math> has a [[finitely presented group|finite presentation]] with solvable [[Word problem for groups|word problem]] and [[conjugacy problem]].<ref name=":3" />

=== Properties of the action ===
{{Needs expansion|date=November 2024}}
Let <math>G</math> be a group acting properly cocompactly by isometries on a CAT(0) space <math>X</math>.

* Any finite subgroup of <math>G</math> fixes a nonempty closed convex set.
* For any infinite order element <math>g\in G</math>, the set <math>\min(g)</math> of elements <math>x\in X</math> such that <math>d(g\cdot x, x) > 0</math> is minimal is a nonempty, closed, convex, <math>g</math>-invariant subset of <math>X</math>, called the ''minimal set'' of <math>g</math>. Moreover, it splits isometrically as a (l²) [[Product metric|direct product]] <math>\min(g) = A\times \R</math> of a closed convex set <math>A\subset X</math> and a geodesic line, in such a way that <math>g</math> acts trivially on the <math>A</math> factor and by translation on the <math>\R</math> factor. A geodesic line on which <math>g</math> acts by translation is always of the form <math>\{a\}\times \R</math>, <math>a\in A</math>, and is called an ''axis'' of <math>g</math>. Such an element is called ''hyperbolic''.
* The flat torus theorem: any free abelian subgroup <math>\Z^n \cong A \subset G</math> leaves invariant a subspace <math>F\subset X</math> isometric to <math>\R^n</math>, and <math>A</math> acts cocompactly on <math>F</math> (hence the quotient <math>F/A</math> is a [[flat torus]]).<ref name=":1" />
* In certain situations, a splitting of <math>G \cong G_1\times G_2</math> as a cartesian product induces a splitting of the space <math>X\cong X_1\times X_2</math> and of the action.<ref name=":2" />

==References==
<references responsive="0">

</references>{{geometry-stub}}
[[Category:Geometric group theory]]
[[Category:Combinatorial group theory]]