Large‐type Artin groups are systolic

Abstract

We prove that Artin groups from a class containing all large‐type Artin groups are systolic. This provides a concise yet precise description of their geometry. Immediate consequences are new results concerning large‐type Artin groups: biautomaticity; existence of EZ‐boundaries; the Novikov conjecture; descriptions of finitely presented subgroups, of virtually solvable subgroups, and of centralizers of elements; the Burghelea conjecture; existence of low‐dimensional models for classifying spaces for some families of subgroups.

Metric systolicity and two-dimensional Artin groups

“…The main difference between the construction in [HO17a] and the one in this paper is that the former is purely combinatorial, while the current one uses both the metric and combinatorial structure. Thus the method in this paper has more flexibility and applies to a much larger class of Artin groups.…”

“…in [AS83,App84,Pri86,Pei96,Bes99]. In [HO17a] the authors undertake similar path showing that Artin groups of large type are systolic, that is, simplicially non-positively curved. This allowed to prove many new results about such groups.…”

“…This definition may be treated as a metric analogue of the definition of systolic complex (see e.g. [Che00,JŚ06,Hag03,HO17a]). Our main tool for exploring features of metrically systolic complexes is the use of disc diagrams.…”

“…We refer to the next subsection for an intuitive explanation of the construction of the complex, as well as comparison with our previous work on constructing systolic complexes for large-type Artin groups from [HO17a].…”

Solvable Subgroup Theorem for simplicial nonpositive curvature

“…For such groups the Theorem follows from their biautomaticity and the fact that their abelian subgroups are finitely generated (see [HO17,Theorem 2.2] for a short account of the proof).…”