A systolic complex/bridged graph is fit when its (metric) intervals are "not too large". We prove that uniformly locally finite fit systolic complexes have Yu's Property A. In particular, groups acting properly on such complexes have Property A, (equivalently) they are exact, and (equivalently) they are boundary amenable. As applications we show that groups from a class containing all large-type Artin groups, as well as all finitely presented graphical $C(3)$--$T(6)$ small cancellation groups, and finitely presented classical $C(6)$ small cancellation groups are exact. We also provide further examples. Our proof relies on a combinatorial criterion for Property~A due to Špakula and Wright.
1. Introduction 1.1. Main results. Embedding, even just in a coarse way, a metric space into a Hilbert space allows one to draw strong conclusions about the former space using the geometry of the latter one. This applies also to (countable) groups equipped with a suitable metric. In particular, the coarse Baum-Connes conjecture holds for spaces embeddable coarsely into a Hilbert space [Yu00]. In the case of countable groups the coarse Baum-Connes conjecture implies e.g. the strong Novikov conjecture [Roe96,Hig00,STY02].
Guoliang Yu [Yu00] introduced a coarse geometric property-Property A-as a criterion for coarse embeddability into a Hilbert space. This property can be seen as a “nonequivariant”, “coarse”, or “weak” variant of amenability. And, similarly as for amenability, there exist numerous equivalent definitions of various flavours-see Definition 2.2 for one. Not surprisingly, Yu’s Property A became of great interest on its own and has been intensively studied over the last decades; see [BO08,Wil09,NY23] for surveys. For a group H seen as a metric space when equipped with the word metric with respect to a finite generating set, the following properties are equivalent [HR00, Oza00, GK02]:
• H has Yu’s Property A;
• H is exact, that is, the functor C * r (H, □) is exact; • H is boundary amenable, that is, G admits an amenable action on a nonempty compact space. These are only three different characterizations, from three different perspectives, respectively: geometric, analytic, and dynamical. See Section 2.2 for further characterizations of Property A for groups. At the moment there exist only two constructions of finitely generated groups without Yu’s Property A: Gromov random monster groups [Gro03], and graphical small cancellation groups constructed in [Osa20]. Basically, all other groups appearing in the literature are believed to have Property A. However, this has been proven only for a few classes of groups, among them: (Gromov) hyperbolic groups [Ada94], Coxeter groups [DJ99], one relator groups [Gue02], linear groups [GHW05], relatively hyperbolic groups (with appropriate parabolics) [Oza06], CAT(0) cubical groups [CN05, BCG + 09, GN11]), mapping class groups of finite-type surfaces [Kid06,Ham09], some isometry groups of buildings [DS09,Cam09,Léc10], groups acting geometrically on 2-dimensional systolic complexes [HO20], outer automorphism groups of free groups [BGH22], and 2-dimensional Artin groups of hyperbolic type [HH21]. For example, it is still unknown whether groups acting geometrically on CAT(0) spaces have Property A, or are coarsely embeddable into Hilbert spaces-see some related questions in Section 10.
In this article we establish Property A for a broad class of systolic complexes by introducing the notion of fitness. Systolic complexes are flag simplicial complexes whose 1-skeleta are bridged graphs, that is, graphs in which isometric cycles have length 3-see details in Section 2.1. Bridged graphs were first introduced by Soltan and Chepoi [SC83], and Farber and Jamison [FJ87]. Systolic complexes were first studied in [Che00], and they were rediscovered by Januszkiewicz and Światkowski [JS06], and Haglund [Hag03] within the framework of Geometric Group Theory. A group is systolic if it acts geometrically, that is, properly and cocompactly, on a systolic complex. Systolicity is a form of combinatorial nonpositive curvature, and providing a systolic structure for a group allows one to grasp a fair control over its geometry. The theory has been successfully used for constructing new exotic examples of groups [JS03,JS06], and for equipping “classical” groups with a nonpositive curvature structure, thus allowing to prove new results about them [HO18,OP18,FO20]. The following main result of the current article extends this latter line of research. We introduce the subclass of fit systolic complexes-those where level sets of intervals are quasi-segments (see Section 3 for details).
Main Theorem. Uniformly locally finite fit systolic complexes have Yu’s Property A.
Below we present our main applications of the Main Theorem-to Artin groups (Corollary A), and to small cancellation groups (Corollaries B & C). We also present a 3dimensional pseudomanifold example (Corollary D). Besides them, the theorem can be used to reprove Property A in other cases: for 2-dimensional systolic complexes (see Remark 3.9) and for few smaller classes of complexes. We believe that the Main Theorem, as well as our general approach (see e.g. Remark 3.10) will be useful for other large classes of groups and complexes; cf. Section 10.
Along the way we prove a result on general bridged graphs/systolic complexes that we believe is of independent interest-Proposition 4.2. We also provide a characterization of fit systolic complexes (Proposition 3.12) and a useful criterion for fitness (Corollary 3.18). We use them in our proofs and we believe that they will be useful in future studies.
Large-type Artin groups.
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