Parabolic subgroups of two‐dimensional Artin groups and systolic‐by‐function complexes

Abstract

We extend previous results by Cumplido, Martin and Vaskou on parabolic subgroups of large-type Artin groups to a broader family of two-dimensional Artin groups. In particular, we prove that an arbitrary intersection of parabolic subgroups of a (2,2)-free twodimensional Artin group is itself a parabolic subgroup.An Artin group is (2,2)-free if its defining graph does not have two consecutive edges labeled by 2. As a consequence of this result, we solve the conjugacy stability problem for this family by applying an algorithm introduced by Cumplido. All of this is accomplished by considering systolic-by-function complexes, which generalize systolic complexes. Systolic-by-function complexes have a more flexible structure than systolic complexes since we allow the edges to have different lengths. At the same time, their geometry is rigid enough to satisfy an analogue of the Cartan-Hadamard theorem and other geometric properties similar to those of systolic complexes.

The Conjugacy Stability Problem for Parabolic Subgroups in Artin Groups

“…Remark 4. After the first preprint of this paper, [3] generalised the results in Cumplido et al [10] and showed that the intersection of parabolic subgroups is a parabolic subgroup for two-dimensional Artin groups with a Coxeter graph-see next section-in which every vertex is disconnected from at most one other vertex. This completed the set of three hypotheses needed in Theorem A and allowed him two apply Algorithm 4 of Sect.…”

Intersection of parabolic subgroups in even Artin groups of FC-type

“…1. if G Γ is of spherical type (see [4]), 2. if G Γ is of FC-type and P 1 is of spherical type (see [11] which generalizes [12] where the result was obtained when both P 1 and P 2 are of spherical type), 3. if G Γ is of large type, that is m({u, v}) ≥ 3 for all {u, v} ∈ E (see [5]), 4.…”

“…if G Γ is a (2,2)-free two-dimensional Artin group, i.e. Γ does not have two consecutive edges labelled by 2 and the geometric dimension of G Γ is two [3],…”

Parabolic subgroups of large-type Artin groups

“…Following the release of this paper, Blufstein generalised this approach to a larger class of two-dimensional Artin groups [ 2 ].…”