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By Bogert M.T., Chertcoff M.

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**Extra resources for A New Group of Dyes from Poison Gases through the 2-Aminothiazoles as Intermediates The Preparation of Thiazole Dyes of Doebner Violet Type**

**Example text**

GUIDO’S BOOK OF CONJECTURES Anantharaman-Delaroche and Renault 19, Higson and Roe ner and Kaminker 21, and Ozawa 22. It is natural to state the following conjecture. 1. A uniformly discrete bounded geometry metric space X has property A if and only if the uniform Roe algebra Cu∗ (X) is exact. In evidence for the conjecture we offer the following. The conjecture is true for any countable discrete group equipped with its natural coarse structure. It is then an easy exercise to show that the conjecture holds for any metric space which admits a proper co-compact action by a group of isometries.

2) We now introduce L2 -Betti numbers βi (6) into this situation. Relevant facts here include that for a finite 2-complex X, (2) (2) (2) χ(X) = χ(2) (X) = β0 (X) − β1 (X) + β2 (X) (2) (2) and βi (X) = βi (π1 (X)) for i = 0, 1. ) (2) A finite 2-complex X is aspherical if χ(X) = 0 and β1 (X) = 0 (3). From (2), the L2 -Betti number condition is satisfied if, for instance, π1 (X) has an infinite subgroup that is (i) amenable and ascendant; or (ii) finitely generated, subnormal and of infinite index.

Thus every automorphism of AΓ lifts to an automorphism of the free group on the vertices of Γ, but the natural map from Out(AΓ ) to GL(n, Z) is not usually surjective. There is a CAT(0) cube complex associated to any RAAG, whose 1skeleton is the Cayley graph of the group, and which has a k-dimensional cube whenever the 1-skeleton of the cube appears (kind of a “cube-flag” complex). The RAAG acts freely on this; the quotient has a loop for each generator and a k-torus for each complete subgraph on k vertices in Γ.

### A New Group of Dyes from Poison Gases through the 2-Aminothiazoles as Intermediates The Preparation of Thiazole Dyes of Doebner Violet Type by Bogert M.T., Chertcoff M.

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