Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Bilipschitz embeddings and bilipschitz equivalences of metric spaces
Definition
Let and be metric spaces and let .
The map is a bilipschitz embedding if there is a real such that
It is a bilipschitz equivalence if it is in addition bijective (Injection, surjection, bijection).
Two metric spaces are bilipschitz equivalent if some bilipschitz equivalence between them exists.
Depends on
Used by
- A bijective quasi-isometry between word metric spaces of finitely generated groups is a bilipschitz equivalence Corollary
- The inclusion of ℤ in ℝ is a quasi-isometry that is neither surjective nor a bilipschitz equivalence Example
- The infinite dihedral group is quasi-isometric to ℤ, and to ℤ×ℤ/2 Example
- The word metrics of ℤ for {1} and for {2,3} differ at 1 and are bilipschitz equivalent Example
- 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 identity map between the word metrics of two finite generating sets is a bilipschitz equivalence 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
10 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. Loh, Geometric Group Theory: An Introduction (2015 course version), Section 5.1 (standard reference, not scraped)
- C. Drutu and M. Kapovich, Geometric Group Theory, Section 8.1 (standard reference, not scraped)