Related papers: Groupes de Garside
We show that the membership problem in a finitely generated submonoid of a graph group (also called a right-angled Artin group or a free partially commutative group) is decidable if and only if the independence graph (commutation graph) is…
We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice,…
We obtain new presentations for the imprimitive complex reflection groups of type $(de,e,r)$ and their braid groups $B(de,e,r)$ for $d,r \ge 2$. Diagrams for these presentations are proposed. The presentations have much in common with…
A planar pure braid consists of $n$ descending smooth arcs, each connecting a point on one horizontal line $\ell_{1}$ to a point on a horizontal line $\ell_{2}$, which is required to be directly below the first point. Two arcs are allowed…
A new family of groups, called trickle groups, is presented. These groups generalize right-angled Artin and Coxeter groups, as well as cactus groups. A trickle group is defined by a presentation with relations of the form $xy = zx$ and…
We define and give axioms for Garside and locally Garside categories. We give an application to Coxeter and Artin groups and Deligne-Lusztig varieties.
We will study the presentations of fundamental groups of the complement of complexified real affine line arrangements that do not contain two parallel lines. By Yoshinaga's minimal presentation, we can give positive homogeneous…
The space of unordered configurations of distinct points in the plane is aspherical, with Artin's braid group as its fundamental group. Remarkably enough, the space of ordered configurations of distinct points on the real projective line,…
We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method…
Recently, Marin and Gonz\'alez-Meneses introduced a class of ``parabolic'' subgroups for generalized braid groups associated to arbitrary complex reflection groups. Using notably Garside group structures on these generalized braid groups,…
We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph…
In this paper we introduce the concept of a Cayley graph automatic group (CGA group or graph automatic group, for short) which generalizes the standard notion of an automatic group. Like the usual automatic groups graph automatic ones enjoy…
From a group $H$ and a non-trivial element $h$ of $H$, we define a representation $\rho: B_n \to \Aut(G)$, where $B_n$ denotes the braid group on $n$ strands, and $G$ denotes the free product of $n$ copies of $H$. Such a representation…
We provide algorithms for performing computations in generalized numerical semigroups, that is, submonoids of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. These semigroups are affine semigroups, which in particular implies…
Let $G$ be a Garside group with Garside element $\Delta$, and let $\Delta^m$ be the minimal positive central power of $\Delta$. An element $g\in G$ is said to be 'periodic' if some power of it is a power of $\Delta$. In this paper, we study…
It is known that Garside groups are strongly translation discrete. In this paper, we show that the translation numbers in a Garside group are rational with uniformly bounded denominators and can be computed in finite time. As an…
M. Picantin introduced the notion of Garside groups of spindle type, generalizing the 3-strand braid group. We show that, for linear Garside groups of spindle type, a normal form and a solution to the conjugacy problem are logspace…
For every group genetic code with finite number of generating and at most with one defining relation we introduce the braid group of this genetic code. This construction includes the braid group of Euclidean plane, the braid groups of…
We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure.…
We study the geometry of algebraic monoids. We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid. Moreover, if this group is reductive, then the monoid is affine. We…