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.
Bounded distance is an equivalence relation and is preserved by pre-composition and by post-composition with a coarse Lipschitz map
Statement
Bounded distance is an equivalence relation and is preserved by pre-composition and by post-composition with a coarse Lipschitz map.
Facts & Assumptions
Given: The hypotheses of the Statement.
Two maps into a metric space are at bounded distance when the distance between their values is bounded uniformly (Bounded distance between two maps into a metric space).
A map is -coarse Lipschitz when , and an -quasi-isometric embedding when in addition (Coarse Lipschitz maps and quasi-isometric embeddings).
Proof
Reflexivity, symmetry and transitivity follow from the metric axioms with the bounds added.
Pre-composition changes no value, so it preserves the bound exactly.
Post-composition with an -coarse Lipschitz map multiplies the bound by and adds ; without the coarse Lipschitz hypothesis the bound need not survive.
Depends on
Used by
- The quasi-isometry group of a metric space Definition
- A map at bounded distance from a quasi-isometric embedding is one, with the additive constant enlarged Lemma
- Quasi-isometries modulo bounded distance form a group, and a quasi-isometry induces an isomorphism of these groups Proposition
- Being quasi-isometric is reflexive, symmetric and transitive Theorem
Dependency tree · two levels
4 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), 264 pp. (standard reference, not scraped)
- C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)