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The quasi-isometry group of a metric space
Definition
Let be a metric space. Consider the set of all quasi-isometries . Two such maps are identified when they are at bounded distance, an equivalence relation by Bounded distance is an equivalence relation and is preserved by pre-composition and by post-composition with a coarse Lipschitz map.
The set of these equivalence classes is written
It is called the quasi-isometry group of . The fact that composition of maps descends to these equivalence classes and makes into a group is proved in Quasi-isometries modulo bounded distance form a group, and a quasi-isometry induces an isomorphism of these groups.
Depends on
Used by
- A nonempty metric space of finite diameter has trivial quasi-isometry group Example
- Scaling maps embed the multiplicative group of nonzero reals into the quasi-isometry group of ℤ Example
- Quasi-isometries modulo bounded distance form a group, and a quasi-isometry induces an isomorphism of these groups Proposition
Dependency tree · two levels
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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)