| Résume | A subspace S of a topological space X is said to separate it if X-S contains more than one path-component. The classical Alexander duality theorem implies that if a subset A of the n-dimensional sphere separates it, then A must be of dimension n-1. 'Coarse separation' is an analogue of topological separation in the world of metric spaces. Coarse separation arises naturally in geometric group theory. Suppose a finitely generated group G (equipped with a word metric) splits over a subgroup C, then C coarsely separates G. I will start by introducing the definition of coarse separation. Given a metric space X, the task is to find a necessary condition that every separating subset of X must satisfy. I will discuss two tools : Growth and Asymptotic dimension, which allow us to obtain such results. I will define everything and give many examples. |
