Angled Coxeter Group Research Articles

3XOR Games with Perfect Commuting Operator Strategies Have Perfect Tensor Product Strategies and are Decidable in Polynomial Time

We consider 3XOR games with perfect commuting operator strategies. Given any 3XOR game, we show existence of a perfect commuting operator strategy for the game can be decided in polynomial time. Previously this problem was not known to be decidable. Our proof leads to a construction, showing a 3XOR game has a perfect commuting operator strategy iff it has a perfect tensor product strategy using a 3 qubit (8 dimensional) GHZ state. This shows that for perfect 3XOR games the advantage of a quantum strategy over a classical strategy (defined by the quantum-classical bias ratio) is bounded. This is in contrast to the general 3XOR case where the optimal quantum strategies can require high dimensional states and there is no bound on the quantum advantage. To prove these results, we first show equivalence between deciding the value of an XOR game and solving an instance of the subgroup membership problem on a class of right angled Coxeter groups. We then show, in a proof that consumes most of this paper, that the instances of this problem corresponding to 3XOR games can be solved in polynomial time.

Some remarks on twin groups

The twin group [Formula: see text] is a right angled Coxeter group generated by [Formula: see text] involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this paper, we study some properties of twin groups whose analogues are well known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups [Formula: see text] have [Formula: see text]-property and are not co-Hopfian for [Formula: see text].

Subgroup graph methods for presentations of finitely generated groups and the contractibility of associated simplicial complexes

In this article we generalize the theory of subgroup graphs of subgroups of free groups, developed by I. Kapovich and A. Myasnikov, based on a work by J. Stallings, to finite index subgroups of finitely generated groups. Given a presentation of a finitely generated group G=〈X|R〉 and a finite connected X-regular graph Γ which fulfills the relators R, we associate to Γ a finite index subgroup H of G. Conversely, the Schreier coset graph of H with respect to X and G is such a graph Γ. Firstly, we study and prove various properties of H in relation to the graph Γ. Secondly, we prove that for many finitely generated infinite groups the order and nerve complexes that we associate to G are contractible. In particular, this is the case for free and free abelian groups, Fuchsian groups of genus g≥2, infinite right angled Coxeter groups, Artin and pure braid groups, infinite virtually cyclic groups, Baumslag–Solitar groups as well as the (free) product of at least two of these, and all finite index subgroups of these groups.

Gromov positive scalar curvature conjecture and rationally inessential macroscopically large manifolds

We give the first examples of rationally inessential but macroscopically large manifolds. Our manifolds are counterexamples to the Dranishnikov rationality conjecture. For some of them we prove that they do not admit a metric of positive scalar curvature, thus satisfy the Gromov positive scalar curvature conjecture. Fundamental groups of our manifolds are finite index subgroups of right angled Coxeter groups. The construction uses small covers of convex polyhedrons (or alternatively Davis complexes) and surgery.

On conjugacy separability of graph products of groups

We show that the class of C-hereditarily conjugacy separable groups is closed under taking arbitrary graph products whenever the class C is an extension closed variety of finite groups. As a consequence we show that the class of C-conjugacy separable groups is closed under taking arbitrary graph products. In particular, we show that right angled Coxeter groups are hereditarily conjugacy separable and 2-hereditarily conjugacy separable, and we show that infinitely generated right angled Artin groups are hereditarily conjugacy separable and p-hereditarily conjugacy separable for every prime number p.

Random graph products of finite groups are rational duality groups

Given an edge-independent random graph G(n,p), we determine various facts about the cohomology of graph products of groups for the graph G(n,p). In particular, the random graph product of a sequence of finite groups is a rational duality group with probability tending to 1 as n goes to infinity. This includes random right angled Coxeter groups as a special case.

CO-CONTRACTIONS OF GRAPHS AND RIGHT-ANGLED COXETER GROUPS

We prove that if b is a co-contraction of , then the right- angled Coxeter group C(b embeds into C(). Further, we provide a graph without an induced long cycle while C() does not contain a hyperbolic surface group.

Coxeter Groups act on CAT(0) cube complexes

We show that any finitely generated Coxeter group acts properly discontinuously on a locally finite, finite dimensional CAT(0) cube complex. For any word hyperbolic or right angled Coxeter group we prove that the cubing is cocompact. We show how the local structure of the cubing is related to the partial order studied by Brink and Howlett in their proof of automaticity for Coxeter groups.