Parabolic Subgroups of Artin Groups of Type Fc
نویسندگان
چکیده
The group AS is called an Artin group and relations sts . . . } {{ } ms,t terms = tst . . . } {{ } ms,t terms are called braid relations. For instance, if S = {s1, . . . , sn} with msi,sj = 3 for |i − j| = 1 and msi,sj = 2 otherwise, then the associated Artin group is the braid group. We denote by A+S the submonoid of AS generated by S. This monoid A+S has the same presentation as the group AS , considered as a monoid presentation ([11]). When we add relations s2 = 1 to the presentation of AS we obtain the Coxeter group WS associated to AS . We say that AS is spherical if WS is finite. The matrix M may be represented by a graph denoted by ΓS , whose set of vertices is S and where an edge joins two vertices if ms,t ≥ 3; these edges are labelled by ms,t if ms,t ≥ 4. We say that AS (or simply S) is indecomposable if the graph ΓS is connected. A subgroup AT of AS generated by a part T of S is called a standard parabolic subgroup, and a subgroup of AS conjugate to a standard parabolic subgroup is called a parabolic subgroup. Van Der Lek showed ([14]) that (AT , T ) is canonically isomorphic to the Artin-Tits system associated to the matrix (ms,t)s,t∈T ; its graph ΓT is the full subgraph of ΓS generated by T . The
منابع مشابه
Pregarside Monoids and Groups, Parabolicity, Amalgamation, and FC Property
We define the notion of preGarside group slightly lightening the definition of Garside group so that all Artin-Tits groups are preGarside groups. This paper intends to give a first basic study on these groups. Firstly, we introduce the notion of parabolic subgroup, we prove that any preGarside group has a (partial) complemented presentation, and we characterize the parbolic subgroups in terms o...
متن کامل2 9 O ct 2 00 6 RELATIVE HYPERBOLICITY AND ARTIN GROUPS
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 2-dimensional Artin group the Deligne complex is Gromov hyperbolic preci...
متن کاملRelative Hyperbolicity and Artin Groups
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 2-dimensional Artin group the Deligne complex is Gromov hyperbolic preci...
متن کاملParabolic subgroups of Garside groups
A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains the Artin-Tits groups of spherical type. We generalise the well-known notion of a parabolic subgroup of an Artin-Tits group into that of a parabolic subgroup of a Garside group. We also define the more general notion...
متن کاملA Note on Relative Hyperbolicity and Artin Groups
In [12], I. Kapovich and P. Schupp showed that certain 2-dimensional Artin groups (those with all relator indices at least 7) are hyperbolic relative (in the sense of Farb) to their non-free rank 2 parabolic subgroups. This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of the associated Deligne complex. We prove a relative version of the Mi...
متن کامل