A new upper bound for the Asymptotic Dimension of RACGs

Abstract

Let W Γ be the Right-Angled Coxeter group with defining graph Γ. We show that the asymptotic dimension of W Γ is smaller than or equal to dim CC (Γ), the clique-connected dimension of the graph. As a corollary we show that W Γ is virtually free if and only if dim CC (Γ) = 1.Theorem 1.2. Let W Γ be the Right-Angled Coxeter group with defining graph Γ. ThenAs a corollary of theorem 1.1 we prove that:Interestingly, we can deduce whether W Γ is virtually free from the cliqueconnected dimension of its defining graph. In particular, we show the following:Proposition 1.4. Let W Γ be the Right-Angled Coxeter group with connected defining graph Γ. Then W Γ is virtually free if and only if dim CC (Γ) = 1.The paper is organized as follows. In subsection 3.1 we start with some basic definitions and some preliminary results that are used in the rest of the paper. Subsection 3.2 is containing some important lemmas, for example, we show that dim CC ( * ) is "monotone" in the following sense: if Γ ′ is a full-subgraphs of Γ, then dim CC (Γ ′ ) ≤ dim CC (Γ). In subsection 3.3, we prove that the clique-connected dimension is increasing in some cases. In subsection 3.4, we prove the main theorem of the paper. In the last subsection, we present some corollaries of the main theorem.Acknowledgments: I would like to thank Panos Papasoglu for his valuable advices during the development of this research work.