Alphabeta MathHow statement and proof provenance work
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Groups acting geometrically on the same space are quasi-isometric
Statement
If two groups act geometrically on the same nonempty geodesic metric space, then they are quasi-isometric.
Facts & Assumptions
Given: Geometric actions of groups and on the same nonempty geodesic metric space .
Under a geometric action on a geodesic metric space, every orbit map is a quasi-isometry (The Svarc-Milnor lemma).
Quasi-isometry is an equivalence relation on metric spaces (Being quasi-isometric is reflexive, symmetric and transitive).
Proof
Choose points . By [L1], the orbit maps based at and make the groups and each quasi-isometric to the common space .
By transitivity of quasi-isometry from [L2], the spaces and are quasi-isometric to each other.
Depends on
Used by
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Sources
- C. Löh, Geometric Group Theory, Sections 4.4 and 5.1 (standard reference, not scraped)
- C. Drutu and M. Kapovich, Lectures on Geometric Group Theory, Chapter 5 (standard reference, not scraped)