Relative hyperbolicity and Artin groups |
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Authors: | Ruth Charney John Crisp |
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Institution: | (1) Department of Mathematics, Brandeis University, Waltham, MA 02454, USA;(2) I.M.B.(UMR 5584 du CNRS), Université de Bourgogne, B.P. 47 870, 21078 Dijon, France |
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Abstract: | This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated
Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups
if and only if its Deligne complex is a Gromov hyperbolic space. For a two-dimensional Artin group the Deligne complex is
Gromov hyperbolic precisely when the corresponding Davis complex is Gromov hyperbolic, that is, precisely when the underlying
Coxeter group is a hyperbolic group. For Artin groups of FC type we give a sufficient condition for hyperbolicity of the Deligne
complex which applies to a large class of these groups for which the underlying Coxeter group is hyperbolic. The key tool
in the proof is an extension of the Milnor-Svarc Lemma which states that if a group G admits a discontinuous, co-compact action by isometries on a Gromov hyperbolic metric space, then G is weakly hyperbolic relative to the isotropy subgroups of the action.
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Keywords: | Relative hyperbolicity Artin group Deligne complex |
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