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A nonempty metric space of finite diameter has trivial quasi-isometry group
Example
A nonempty metric space of finite diameter has trivial quasi-isometry group.
Facts & Assumptions
Given: The objects and hypotheses in the Example.
The quasi-isometry group of a metric space is the set of quasi-isometries of it modulo bounded distance (The quasi-isometry group of a metric space).
The quasi-isometry group is a group under composition, and a quasi-isometry induces an isomorphism between the quasi-isometry groups of its source and target (Quasi-isometries modulo bounded distance form a group, and a quasi-isometry induces an isomorphism of these groups).
Bounded subset. is bounded if or there are and a real with . (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).
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).
Verification
In a space of finite diameter every self-map is at distance at most the diameter from the identity.
So there is exactly one bounded-distance class, and the quasi-isometry group is trivial.
Depends on
- The quasi-isometry group of a metric space
- Quasi-isometries modulo bounded distance form a group, and a quasi-isometry induces an isomorphism of these groups
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
- Bounded distance between two maps into a metric space
Used by
Nothing in the library uses this result yet.
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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)