In this paper, we study the minimality of the boundary of a Coxeter system. We show that for a Coxeter system (W,S) if there exist a maximal spherical subset T of S and an element s0 ∈ S such that m(s0, t) ≥ 3 for each t ∈ T and m(s0, t0) = ∞ for some t0 ∈ T , then every orbit Wα is dense in the boundary ∂Σ(W,S) of the Coxeter system (W,S), hence ∂Σ(W,S) is minimal, where m(s0, t) is the order ...

Let (W, S) be a Coxeter system, with W a finite, irreducible Coxeter group and let GW be the associated Artin group (see Bourbaki [5] for an introduction to Coxeter groups and their classifications and Brieskorn and Saito [6] for relations between Coxeter groups and Artin groups ). The main objects of study of this paper are the Artin groups of type An . We recall that the Artin group GAn is th...

Let (W,S) be a Coxeter system, let S = I ∪ J be a partition of S such that no element of I is conjugate to an element of J , let J̃ be the set of WI -conjugates of elements of J and let W̃ be the subgroup of W generated by J̃ . We show that W = W̃ ⋊WI and that J̃ is the canonical set of Coxeter generators of the reflection subgroup W̃ of W . We also provide algebraic and geometric conditions for an e...

A solution of the isomorphism problem is presented for the class of Coxeter groups W that have a finite set of Coxeter generators S such that the underlying graph of the presentation diagram of the system (W, S) has the property that every cycle of length at least four has a cord. As an application , we construct counterexamples to two main conjectures concerning the isomorphism problem for Cox...

Let (W, S, Γ) be an irreducible finitely presented Coxeter system. The present paper is mainly concerned with conjugacy relation on Coxeter elements in the case where Γ containing just one circle, in particular when Γ is itself a circle. In the cases where Γ is either a three multiple circle or a circle with three nodes, we show that the ss-equivalence relation on Coxeter elements of W is the s...

The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n− 1)-dimensional spheres. The number of these spheres is the boole...

We establish the existence of an IC basis for the generalized Temperley{ Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra is nite dimensional.

approved: William A. Bogley A Coxeter group W is said to be rigid if, given any two Coxeter systems (W,S) and (W,S ′), there is an automorphism ρ of W which carries S to S ′ In this dissertation we review rigidity results of D. Radcliffe [29] and of R. Charney and M. Davis [11], noting that certain restrictions must be placed on the Coxeter group in order to obtain rigidity. Radcliffe examined ...

In this paper, we give a class of reflection rigid Coxeter systems. Let (W, S) be a Coxeter system. Suppose that (1) for each s, t ∈ S such that m(s, t) is odd, {s, t} is a maximal spherical subset of S, (2) there does not exist a three-points subset {s, t, u} ⊂ S such that m(s, t) and m(t, u) are odd, and (3) for each s, t ∈ S such that m(s, t) is odd, the number of maximal spherical subsets o...

A Coxeter system is an ordered pair (W,S) where S is the generating set in a particular type of presentation for the Coxeter group W . A subgroup of W is called special if it is generated by a subset of S. Amalgamated product decompositions of a Coxeter group having special factors and special amalgamated subgroup are easily recognized from the presentation of the Coxeter group. If a Coxeter gr...