The cyclically reduced product of two words is the cyclically reduced form of the concatenation of the two words. While the reduced form of such a concatenation (which is the product of the free group) verifies many basic properties like for example associativity, the same is not true for the cyclically reduced product which has been very little studied in the literature. Recently Sergei Ivanov has proved that the Andrews-Curtis conjecture (stated in 1965 and still not solved) is equivalent to a formulation where the reduced product is replaced by the cyclically reduced product (and the conjugations replaced by cyclic permutations). In this paper we study properties of the cyclically reduced product $*$ and of the set of cyclically reduced words $\hat{\mathcal{F}(X)}$ equipped with $*$. In particular we find that even if $*$ is not commutative nor verifies the Latin square property, weaker versions of these properties hold true. We also show that $\hat{\mathcal{F}(X)}$ equipped with $*$ and with cyclic permutations enjoys similar properties as the free group equipped with the reduced product and conjugations.
The cyclically reduced product of two words $u, v$, denoted $u * v$, is the cyclically reduced form of the concatenation of $u$ by $v$. This product is not associative. Recently S. V. Ivanov has proved that the Andrews-Curtis conjecture can be restated in terms of the cyclically reduced product and cyclic permutations instead of the reduced product and conjugations. In a previous paper we have started a thorough study of $*$ and of the structure of the set of cyclically reduced words $\hat{\mathcal{F}}(X)$ equipped with $*$. In particular we have found that a certain number of properties of th…
The cyclically reduced product of two words $u, v$, denoted $u * v$, is the cyclically reduced form of the concatenation of $u$ by $v$. This product is not associative. Recently S. V. Ivanov has proved that the Andrews-Curtis conjecture can be restated in terms of the cyclically reduced product and cyclic permutations instead of the reduced product and conjugations. In a previous paper we have proved that $*$ verifies generalizations of properties of the product in the free group. In another previous paper we have proved that $*$ verifies a generalized version of the associativity property in …
Let $W$ be a Coxeter group. We provide a precise description of the conjugacy classes in $W$, in the spirit of Matsumoto's theorem. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of $W$, thereby proving a conjecture of A. Cohen from 1994.
For an infinite Coxeter system, one can extend the weak right order to the set of infinite reduced words. This is called limit weak order. In [Transformation Groups 18(1), 2013, 179-231], Lam and Pylyavskyy showed that for affine Weyl groups of type $\widetilde{A}_n$ minimal infinite reduced words under the limit weak order are precisely those infinite Coxeter elements and asked the question of characterization, in terms of infinite reduced words, of the minimal elements of the limit weak order for other affine types. In this paper, we answer this question by characterizing the minimal infinit…
In this paper, we establish a bijection between the infinite reduced words of an affine Weyl group and certain biclosed sets of its positive system and determine all finitely generated biclosed sets in the positive system of an affine Weyl group. Using these results, we show first that the biclosed sets in the standard positive system of rank 3 affine Weyl groups when ordered by inclusion form a complete algebraic ortholattice and secondly that the (generalized) braid graphs of those Coxeter groups are connected, which can be thought of as an infinite version of Tit's solution to the word prob…
