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Actions of Right-Angled Coxeter Groups on the Croke Kleiner Spaces  [PDF]
Yulan Qing
Mathematics , 2014,
Abstract: It is an open question whether right-angled Coxeter groups have unique group-equivariant visual boundaries. Croke and Kleiner present a right-angled Artin group with more than one visual boundary. In this paper we present a right-angled Coxeter group with non-unique equivariant visual boundary. The main theorem is that if right-angled Coxeter groups act geometrically on a Croke-Kleiner spaces, then the local angles in those spaces all have to be right angles. We present a speci?c right-angled Coxeter group with non-unique equivariant visual boundary. However, we conjecture that the right an- gled Coxeter groups that can act geometrically on a given CAT(0) space are far from unique.
Geodesic growth in right-angled and even Coxeter groups  [PDF]
Yago Antolín,Laura Ciobanu
Mathematics , 2012,
Abstract: The objective of this paper is to detect which combinatorial properties of a regular graph can completely determine the geodesic growth of the right-angled Coxeter or Artin group this graph defines, and to provide the first examples of right-angled and even Coxeter groups with the same geodesic growth series.
Braid groups and right angled Artin groups  [PDF]
Frank Connolly,Margaret Doig
Mathematics , 2004,
Abstract: The n-string braid group of a graph X is defined as the fundamental group of the n-point configuration space of the space X. This configuration space is a finite dimensional aspherical space. A. Abrams and R. Ghrist have conjectured that this braid group is a right angled Artin group if X is planar. We prove their conjecture if X is a tree whose nodes all lie in a single interval of X.
Embeddings of right-angled Artin groups  [PDF]
Travis Scrimshaw
Mathematics , 2010,
Abstract: We explicitly construct an embedding of a right-angled Artin group into a classical pure braid group. Using this we obtain a number of corollaries describing embeddings of arbitrary Artin groups into right-angled Artin groups and linearly independent subgroups of a right-angled Artin group.
The action dimension of right-angled Artin groups  [PDF]
Grigori Avramidi,Michael W. Davis,Boris Okun,Kevin Schreve
Mathematics , 2014,
Abstract: The action dimension of a discrete group $\Gamma$ is the smallest dimension of a contractible manifold which admits a proper action of $\Gamma$. Associated to any flag complex $L$ there is a right-angled Artin group, $A_L$. We compute the action dimension of $A_L$ for many $L$. Our calculations come close to confirming the conjecture that if an $\ell^2$-Betti number of $A_L$ in degree $l$ is nonzero, then the action dimension of $A_L$ is $\ge 2l$.
Abelian splittings of Right-Angled Artin Groups  [PDF]
Daniel Groves,Michael Hull
Mathematics , 2015,
Abstract: We characterize when (and how) a Right-Angled Artin group splits nontrivially over an abelian subgroup.
Rigidity of Right-Angled Coxeter Groups  [PDF]
David G. Radcliffe
Mathematics , 1999,
Abstract: If S and S' are two finite sets of Coxeter generators for a right-angled Coxeter group W, then the Coxeter systems (W,S) and (W,S') are equivalent.
Divergence in right-angled Coxeter groups  [PDF]
Pallavi Dani,Anne Thomas
Mathematics , 2012,
Abstract: Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex.
Topology of Random Right Angled Artin Groups  [PDF]
Armindo Costa,Michael Farber
Mathematics , 2009,
Abstract: In this paper we study topological invariants of a class of random groups. Namely, we study right angled Artin groups associated to random graphs and investigate their Betti numbers, cohomological dimension and topological complexity. The latter is a numerical homotopy invariant reflecting complexity of motion planning algorithms in robotics. We show that the topological complexity of a random right angled Artin group assumes, with probability tending to one, at most three values. We use a result of Cohen and Pruidze which expresses the topological complexity of right angled Artin groups in combinatorial terms. Our proof deals with the existence of bi-cliques in random graphs.
On the arboreal structure of right-angled Artin groups  [PDF]
?erban A. Basarab
Mathematics , 2009,
Abstract: The present article continues the study of median groups initiated in [6, 9, 10]. Some classes of median groups are introduced and investigated with a stress upon the class of the so called A-groups which contains as remarkable subclasses the lattice ordered groups and the right-angled Artin groups. Some general results concerning A-groups are applied to a systematic study of the arboreal structure of right-angled Artin groups. Structure theorems for foldings, directions, quasidirections and centralizers are proved.
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