Parabolic subgroups of Artin groups of FC type

Abstract

Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed using parabolic subgroups. We extend the construction of the complex of parabolic subgroups to FC type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is δ−hyperbolic.

Lattices, injective metrics and the $K(π,1)$ conjecture

“…The proof of Morris-Wright (see [MW19]) for the intersection of parabolic subgroups in FC type adapts directly to our situation. It relies mainly of the result by Cumplido et al (see [CGGMW19]) that in a spherical type Artin group, the intersection of parabolic subgroups is a parabolic subgroup.…”

“…Godelle studied the centralizer and normalizer of standard parabolic subgroups (see [God07]). Morris-Wright studied the intersections of parabolic subgroups (see [MW19]). It turns out almost all the arguments merely used the existence of an equivariant geodesic bicombing on the Deligne complex (see the end of Section 3 for a definition of a bicombing).…”

“…If an Artin group satisfies Conjecture K, then the K(π, 1) conjecture follows. Moreover, we may also list consequences of work of Crisp (see [Cri00]), Godelle (see [God07]) and Morris-Wright (see [MW19] and [CGGMW19]) that rely on the assumption that the Deligne complex has a CAT(0) metric. However, most of the arguments only use the geodesic bicombing.…”

Curve graphs for Artin-Tits groups of type $B$, $\widetilde A$ and $\widetilde C$ are hyperbolic

“…Accordingly, we will always use the exponent notation for conjugacy in a group G: given g, h ∈ G, h g = g −1 hg. Most of the known properties of the graph C parab (Γ) are gathered in [7] and [17]. For example, if A Γ is irreducible and of spherical type (Γ with at least 3 vertices), then C parab (A Γ ) is connected and has infinite diameter ([17, Lemma 5.2] and [7,Corollary 4.13]).…”

Morse Quasiflats II

“…For many other natural examples (e.g. Coxeter groups, Artin groups or some cubical groups), the definition of Σ X necessarily involves flats (or abelian subgroups) which do not have top rank, in order to avoid substantial loss of information [KK14, MW19,DH17]. This naturally leads to the study of Morse quasiflats.…”