Abstract
We introduce the notion of metrically systolic simplicial complexes. We study geometric and large-scale properties of such complexes and of groups acting on them geometrically. We show that all two-dimensional Artin groups act geometrically on metrically systolic complexes. As direct corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: we prove that the Conjugacy Problem is solvable, and that the Dehn function is quadratic. We also show several large-scale features of finitely presented subgroups of two-dimensional Artin groups, lying background for further studies concerning their quasi-isometric rigidity.
Highlights
Artin groups are among most intensively studied classes of groups in Geometric Group Theory
In [33] the authors undertake similar path showing that Artin groups of large type are systolic, that is, simplicially non-positively curved
In the current article we exhibit a non-positive-curvature-like structure of all two-dimensional Artin groups and all their finitely presented subgroups, and conclude a number of new algorithmic, and large-scale geometric results for those groups
Summary
Artin groups are among most intensively studied classes of groups in Geometric Group Theory. Metric systolicity enables us to stay in the 2-dimensional world—one need to study only CAT(0) disc diagrams This will be convenient for our further work in [34] concerning quasi-isometries of 2-dimensional Artin groups. Solvability of the Conjugacy Problem for 2-dimensional Artin groups and their finitely presented subgroups follows directly from Theorem 1.1 (4). For the proof of the strong form of (3), as presented in Theorem 3.7 in the text, the use of metric systolicity is very convenient This result, in turn, is a crucial ingredient in the proof of the Morse Lemma for two-dimensional quasi-discs (see the proof of Theorem 3.9). The latter is an important large-scale feature of metrically systolic complexes, groups, and of 2-dimensional Artin groups.
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