40 research outputs found
A combinatorial non-positive curvature I: weak systolicity
We introduce the notion of weakly systolic complexes and groups, and initiate
regular studies of them. Those are simplicial complexes with
nonpositive-curvature-like properties and groups acting on them geometrically.
We characterize weakly systolic complexes as simply connected simplicial
complexes satisfying some local combinatorial conditions. We provide several
classes of examples --- in particular systolic groups and CAT(-1) cubical
groups are weakly systolic. We present applications of the theory, concerning
Gromov hyperbolic groups, Coxeter groups and systolic groups.Comment: 35 pages, 1 figur
A construction of hyperbolic Coxeter groups
We give a simple construction of Gromov hyperbolic Coxeter groups of
arbitrarily large virtual cohomological dimension. Our construction provides
new examples of such groups. Using this one can construct e.g. new groups
having some interesting asphericity properties.Comment: 14 page
Ideal boundary of 7-systolic complexes and groups
We prove that ideal boundary of a 7-systolic group is strongly hereditarily
aspherical. For some class of 7-systolic groups we show their boundaries are
connected and without local cut points, thus getting some results concerning
splittings of those groups.Comment: 19 pages. Algebr. Geom. Topol., to appea
Metric systolicity and two-dimensional Artin groups
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.Comment: final preprint version, to appear in Math. An