On Growth Types of Quotients of Coxeter Groups by Parabolic Subgroups
نویسندگان
چکیده
منابع مشابه
On Growth Types of Quotients of Coxeter Groups by Parabolic Subgroups
The principal objects studied in this note are Coxeter groups W that are neither finite nor affine. A well known result of de la Harpe asserts that such groups have exponential growth. We consider quotients of W by its parabolic subgroups and by a certain class of reflection subgroups. We show that these quotients have exponential growth as well. To achieve this, we use a theorem of Dyer to con...
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In this article, we consider infinite, non-affine Coxeter groups. These are known to be of exponential growth. We consider the subsets of minimal length coset representatives of parabolic subgroups and show that these sets also have exponential growth. This is achieved by constructing a reflection subgroup of our Coxeter group which is isomorphic to the universal Coxeter group on three generato...
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Let (W,S) be a Coxeter system, and let X be a subset of S. The subgroup of W generated by X is denoted by WX and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of WX in W is the subgroup of w in W such that wWXw ∩WX has finite index in both WX and wWXw . The subgroup WX can be decomposed in the form ...
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We improve a bound of Borcherds on the virtual cohomological dimension of the non-reflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter subdiagram rather than the number of nodes. A consequence is an extension of Brink’s result that the non-reflection part of a reflection centralizer is fre...
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In this paper, we investigate boundaries of parabolic subgroups of Coxeter groups. Let (W, S) be a Coxeter system and let T be a subset of S such that the parabolic subgroup WT is infinite. Then we show that if a certain set is quasi-dense in W , then W∂Σ(WT , T ) is dense in the boundary ∂Σ(W, S) of the Coxeter system (W, S), where ∂Σ(WT , T ) is the boundary of (WT , T ).
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ژورنال
عنوان ژورنال: Communications in Algebra
سال: 2008
ISSN: 0092-7872,1532-4125
DOI: 10.1080/00927870701724409