Related papers: Linearity of Generalized Cactus Groups

In 2011 Eriko Hironaka introduced an interesting generalization of Coxeter groups, motivated by studying certain mapping classes. The generalization is by labeling the vertices of a Coxeter graph either by +1 or by -1, and then generalizing…

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of…

The cactus group was introduced by Henriques and Kamnitzer as an analogue of the braid group. In this note, we provide an explicit description of the relationship between the pure cactus group of degree three and the configuration space of…

Results are obtained concerning the roots of asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a…

We apply the techniques of symmetric generation to establish the standard presentations of the finite simply laced irreducible finite Coxeter groups, that is the Coxeter groups of types An, Dn and En, and show that these are naturally…

In this paper, we analyze the faithful representations of the dihedral groups, and prove that the Coxeter groups can be determined by the proper joint spectrum of their faithful representations.

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the…

The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$, one can associate an $\R$-tree called the continuous cactus of $E$. We prove under general assumptions…

In this article, we introduce rotation groups as a common generalisation of Coxeter groups and graph products of groups (including right-angled Artin groups). We characterise algebraically these groups by presentations (periagroups) and we…

We consider a particular class of Garside groups, which we call circular groups. We mainly prove that roots are unique up to conjugacy in circular groups. This allows us to completely classify these groups up to isomorphism. As a…

In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse…

An axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras was presented in [1]. Combinatorial aspects of this construction are studied in this paper. In particular, the symmetric group case is…

The Linearization Theorem for proper Lie groupoids organizes and generalizes several results for classic geometries. Despite the various approaches and recent works on the subject, the problem of understanding invariant linearization…

Approximate lattices are aperiodic generalisations of lattices of locally compact groups that were first studied in seminal work of Yves Meyer. They are defined as those uniformly discrete approximate subgroups (symmetric subsets stable…

A class of generalized nonlinear Kolmogorov equations is investigated. We present the group classification of Lie symmetries of the class with respect to the group of equivalence transformations. We find a number of exact solutions of…

In this paper we build an abstract description of vertex algebras from their basic axioms. Starting with Borcherds' notion of a vertex group, we naturally construct a family of multilinear singular maps parameterised by trees. These…

A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a $k$-cactus is a connected graph in which each edge is contained in at most $k$ cycles where $k\ge 1$. It…

We develop a coarse notion of bundle and use it to understand the coarse geometry of group extensions and, more generally, groups acting on proper metric spaces. The results are particularly sharp for groups acting on (locally finite) trees…

In this work, we will introduce the notion of generalized topological groups using generalized topological structure and generalized continuity defined by ?A. Cs?asz?ar [2]. We will discuss some basic properties of this kind of structures…

The notion of a linear Coxeter system introduced by Vinberg generalizes the geometric representation of a Coxeter group. Our main theorem asserts that if $v$ is an element of the Tits cone of a linear Coxeter system and $\cW$ is the…