Search results for "Artin group"
showing 10 items of 23 documents
Automorphisms of 2–dimensional right-angled Artin groups
2007
We study the outer automorphism group of a right-angled Artin group AA in the case where the defining graph A is connected and triangle-free. We give an algebraic description of Out.AA/ in terms of maximal join subgraphs in A and prove that the Tits’ alternative holds for Out.AA/. We construct an analogue of outer space for Out.AA/ and prove that it is finite dimensional, contractible, and has a proper action of Out.AA/. We show that Out.AA/ has finite virtual cohomological dimension, give upper and lower bounds on this dimension and construct a spine for outer space realizing the most general upper bound. 20F36; 20F65, 20F28
Parabolic Subgroups of Artin Groups
1997
Abstract Let ( A , Σ) be an Artin system. For X ⊆ Σ, we denote by A X the subgroup of A generated by X . Such a group is called a parabolic subgroup of A . We reprove Van der Lek's theorem: “a parabolic subgroup of an Artin group is an Artin group.” We give an algorithm which decides whether two parabolic subgroups of an Artin group are conjugate. Let A be a finite type Artin group, and let A X be a parabolic subgroup with connected associated Coxeter graph. The quasi-centralizer of A X in A is the set of β in A such that β X β −1 = X . We prove that the commensurator of A X in A is equal to the normalizer of A X in A , and that this group is generated by A X and the quasi-centralizer of…
The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group
2000
It was conjectured by Tits that the only relations amongst the squares of the standard generators of an Artin group are the obvious ones, namely that a^2 and b^2 commute if ab=ba appears as one of the Artin relations. In this paper we prove Tits' conjecture for all Artin groups. More generally, we show that, given a number m(s)>1 for each Artin generator s, the only relations amongst the powers s^m(s) of the generators are that a^m(a) and b^m(b) commute if ab=ba appears amongst the Artin relations.
Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups
1999
It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter-type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups. Define a Gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths, and where left and right lowest common multiples exist. A Garside monoid is a Gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition. A Gaussian group is the group of fractions of a Gaussian monoid, and a Garside group is the group of fractions of a Garside monoid. Braid groups and, more genera…
Commensurability classification of a family of right-angled Coxeter groups
2008
We classify the members of an infinite family of right-angled Coxeter groups up to abstract commensurability.
Conjugacy problem for braid groups and Garside groups
2003
We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee. This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among others).
On the Toeplitz algebras of right-angled and finite-type Artin groups
1999
The graph product of a family of groups lies somewhere between their direct and free products, with the graph determining which pairs of groups commute and which do not. We show that the graph product of quasi-lattice ordered groups is quasi-lattice ordered, and, when the underlying groups are amenable, that it satisfies Nica's amenability condition for quasi-lattice orders. As a consequence the Toeplitz algebras of these groups are universal for covariant isometric representations on Hilbert space, and their representations are faithful if the isometries satisfy a properness condition given by Laca and Raeburn. An application of this to right-angled Artin groups gives a uniqueness theorem …
Automorphism groups of some affine and finite type Artin groups
2004
We observe that, for fixed n ≥ 3, each of the Artin groups of finite type An, Bn = Cn, and affine type ˜ An−1 and ˜ Cn−1 is a central extension of a finite index subgroup of the mapping class group of the (n + 2)-punctured sphere. (The centre is trivial in the affine case and infinite cyclic in the finite type cases). Using results of Ivanov and Korkmaz on abstract commensurators of surface mapping class groups we are able to determine the automorphism groups of each member of these four infinite families of Artin groups. A rank n Coxeter matrix is a symmetric n × n matrix M with integer entries mij ∈ N ∪ {∞} where mij ≥ 2 for ij, and mii = 1 for all 1 ≤ i ≤ n. Given any rank n Coxeter matr…
Ordering Garside groups
2017
We introduce a condition on Garside groups that we call Dehornoy structure. An iteration of such a structure leads to a left order on the group. We show conditions for a Garside group to admit a Dehornoy structure, and we apply these criteria to prove that the Artin groups of type A and I2(m), m ≥ 4, have Dehornoy structures. We show that the left orders on the Artin groups of type A obtained from their Dehornoy structures are the Dehornoy orders. In the case of the Artin groups of type I2(m), m ≥ 4, we show that the left orders derived from their Dehornoy structures coincide with the orders obtained from embeddings of the groups into braid groups.
SURFACE SUBGROUPS OF RIGHT-ANGLED ARTIN GROUPS
2007
We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\geq 5$. We construct another eight "forbidden" graphs and show that every graph $K$ on $\le 8$ vertices either contains one of our examples, or contains a hole of length $\ge 5$, or has the property that $A(K)$ does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a \RAAG to contain no hyperbolic surface subgroups. We prove that for one of these "forbidden" subgraphs $P_2(6)$, …