| Resumo: | <p>The asymptotic dimension is a large scale analogue of topological dimension which proved to be a very useful tool in Group Theory and Topology.</p>
<p>In Chapter 1, we prove a new inequality for the asymptotic dimension of HNN-extensions. We deduce that the asymptotic dimension of every finitely generated one relator group is at most two, confirming a conjecture of A.Dranishnikov. As further corollaries, we calculate the asymptotic dimension of RAAGs, and we give a new upper bound for the asymptotic dimension of fundamental groups of graphs of groups.</p>
<p>In Chapter 2, we give a new upper bound for the asymptotic dimension of RACGs. To be more precise, let W<sub>Γ</sub> be the right-angled Coxeter group with defining graph Γ. We show that the asymptotic dimension of W<sub>Γ</sub> is smaller than or equal to dim<sub>CC</sub>(Γ), the clique-connected dimension of the graph.</p>
<p>In Chapter 3, we give some new results for the asymptotic dimension of Artin and Coxeter groups. If AΓ (WΓ) is the Artin (Coxeter) group with defining graph Γ we denote by Sim(Γ) the number of vertices of the largest clique in Γ. We show that asdimA<sub>Γ</sub> ≤ Sim(Γ), if Sim(Γ) = 2. We conjecture that the inequality holds for every Artin group. We prove that if for all free of infinity Artin (Coxeter) groups the conjecture holds, then it holds for all Artin (Coxeter) groups. As a corollary, we show that asdimWΓ ≤ Sim(Γ) for all Coxeter groups, which is the best known upper bound for the asymptotic dimension of Coxeter Groups. As a further corollary, we show that the asymptotic dimension of any large type Artin groups with Sim(Γ) = 3 is at most three.</p>
<p>The fourth chapter is inspired by a theorem of topological dimension theory which states that every topological space X of dimension n can be separated by a subset Y of dimension equal to or smaller than n − 1. We give an analogous theorem in the asymptotic dimension case. To be
more precise, we prove that every geodesic metric space of asymptotic dimension n containing a bi-infinite geodesic can be coarsely separated by a subset S of asymptotic dimension equal to or smaller than n − 1. Moreover, we define asymptotic Cantor manifolds and we state some questions related to them.</p>
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