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Coarsely separation of groups and spaces

Panagiotis Tselekidis

Abstract

Inspired by a classical theorem of topological dimension theory, 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 $n-1$.\\ We define asymptotic Cantor manifolds, and we prove that every finitely generated group contains such a manifold. We also state some questions related to them.

Coarsely separation of groups and spaces

Abstract

Inspired by a classical theorem of topological dimension theory, we prove that every geodesic metric space of asymptotic dimension containing a bi-infinite geodesic can be coarsely separated by a subset of asymptotic dimension equal to or smaller than .\\ We define asymptotic Cantor manifolds, and we prove that every finitely generated group contains such a manifold. We also state some questions related to them.
Paper Structure (6 sections, 17 theorems, 15 equations, 5 figures)

This paper contains 6 sections, 17 theorems, 15 equations, 5 figures.

Table of Contents

  1. Introduction.
  2. Weak Coarse Separation of Spaces.
  3. Coarse Separation of Spaces.
  4. Asymptotic analogues of Aleksandrov's theorem.
  5. Results.
  6. Questions.

Key Result

Theorem 1.1

Let $X$ be a geodesic metric space containing a bi-infinite geodesic. If $asdimX=n>0$, then there exists a subspace $Y$ of asymptotic dimension strictly less than $n$ which coarsely separates $X$.

Figures (5)

  • Figure 1:
  • Figure 2: For $\mathcal{U},V \in \overline{\mathcal{U}}_{S_{d}}$
  • Figure 3:
  • Figure 4: Here we have an illustration of the result of step 3 for n=1.
  • Figure 5: This is the picture we have.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 28 more