Coarse Lipschitz maps and quasi-isometric embeddings

Depends on

• Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric
• Lipschitz map, $\alpha$-Hölder map for rational $0 < \alpha \le 1$, and contraction

Used by

• A bijective quasi-isometry between word metric spaces of finitely generated groups is a bilipschitz equivalence Corollary
• A single map exhibiting a quasi-isometry that is discontinuous, non-injective and non-surjective Counterexample
• Coarsely dense subsets, quasi-inverses and quasi-isometries Definition
• Quasi-geodesics and quasi-geodesic metric spaces Definition
• The inclusion of ℤ in ℝ is a quasi-isometry that is neither surjective nor a bilipschitz equivalence Example
• FALSE: any two infinite finitely generated groups are quasi-isometric False statement
• FALSE: every quasi-isometry is continuous, or bijective False statement
• A map at bounded distance from a quasi-isometric embedding is one, with the additive constant enlarged Lemma
• Bounded distance is an equivalence relation and is preserved by pre-composition and by post-composition with a coarse Lipschitz map Lemma
• Composites of coarse Lipschitz maps and of quasi-isometric embeddings are again such, with explicit constants Lemma
• A coarse Lipschitz map between word metric spaces of finitely generated groups is Lipschitz Proposition
• A subgroup of finite index in a finitely generated group is finitely generated, and its inclusion is a quasi-isometry Proposition
• Every isometry is a bilipschitz equivalence and every bilipschitz equivalence is a quasi-isometry, and two metrics on one set are Lipschitz equivalent exactly when the identity is a bilipschitz equivalence between them Proposition
• The composite of a quasi-geodesic with a quasi-isometric embedding is a quasi-geodesic, with computed constants Proposition
• The nonempty metric spaces quasi-isometric to a one-point space are exactly those of finite diameter Proposition
• The quotient map by a finite normal subgroup is a quasi-isometry of word metric spaces Proposition
• The vertex set of a connected simple graph with its path metric is a (1,1)-quasi-geodesic space Proposition
• A quasi-isometric embedding with coarsely dense image has a quasi-inverse quasi-isometric embedding Theorem
• Two metric spaces are quasi-isometric if and only if each contains a separated net and the two nets are bilipschitz equivalent Theorem

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• C. Drutu and M. Kapovich, Geometric Group Theory, Section 8.1 (standard reference, not scraped)
• C. Loh, Geometric Group Theory: An Introduction (2015 course version), Section 5.1 (standard reference, not scraped)