Three-dimensional FC Artin Groups are CAT(0)

Abstract

Building upon earlier work of T. Brady, we construct locally CAT(0) classifying spaces for those Artin groups which are three-dimensional and which satisfy the FC (flag complex) condition. The approach is to verify the 'link condition' by applying gluing arguments for CAT(1) spaces and by using the curvature testing techniques of Elder and

The 6-strand braid group is CAT(0)

“…He showed that these complexes carry a piecewise Euclidean metric of non-positive curvature and have as fundamental group the Artin groups under consideration. A generalisation of this was proved by Bell [2].…”

“…2b) F is the following edge of rank (1, 2) with dominant vertex v as in (2a) (2c) F is the following edge of rank(1,2) with dominant vertex v as in (2a) (3a) F is a single vertex of rank 3, namely either v = {{1, 2, 3, 5}, {4}, {6}} or v = {{1, 2, 4, 5}, {3}, {6}}(3b) F is the following edge of rank (3, 4) with dominant vertex v as in (3a) Proof. Write v 1 , .…”

Metric systolicity and two-dimensional Artin groups

“…It is so because 2-dimensional Artin groups are torsion-free by [CD95b], and their cyclic subgroups are undistorted (see Theorem 1.4 below). Prior to our result solvability of the Conjugacy Problem was established only for a few particular subclasses of Artin groups: braid groups [Gar69], finite type Artin groups [BS72, Del72, Cha92, Cha95], large-type Artin groups [App84,AS83], triangle-free Artin groups [Pri86], 3-dimensional Artin groups of type FC [Bel05], certain 2-dimensional Artin groups with 3 generators [BC02], some Artin groups of Euclidean type [CC05,Dig06,Dig12,McC15,MS17], RAAG's [Ser89,Van94,HM95,CGW09]. In particular, the question about solvability of the Conjugacy Problem has been open for the class of 2-dimensional Artin groups.…”

Cubulation of some triangle-free Artin groups

“…Concerning CAT(0) spaces, R. Charney asks whether every Artin-Tits group acts properly and cocompactly on a CAT(0) space (see [Cha]). Very few cases are known, essentially right-angled Artin groups (see [CD95]), groups with few generators (see [Bra00], [BM10], [HKS16]) and groups with sufficiently large labels (see [BC02], [BM00], [Bel05], [Hae19]).…”