PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-26

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Quasi-isometries modulo bounded distance form a group, and a quasi-isometry induces an isomorphism of these groups

Statement

Quasi-isometries modulo bounded distance form a group, and a quasi-isometry induces an isomorphism of these groups.

Facts & Assumptions

Given: The hypotheses of the Statement.

[F1]

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).

[L1]

Bounded distance is an equivalence relation, is preserved by pre-composition, and is preserved by post-composition with a coarse Lipschitz map (Bounded distance is an equivalence relation and is preserved by pre-composition and by post-composition with a coarse Lipschitz map).

[L2]

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

[L3]

A group is a monoid $(G, *, e)$ in which every element is invertible. (Group and abelian group).

[L4]

Group isomorphisms, automorphisms and the set $Aut (G)$ . (Group isomorphisms, automorphisms and the set $Aut (G)$ ).

[L5]

A binary relation $\sim$ on $A$ is an equivalence relation when it is reflexive on $A$ , symmetric and transitive, that is, when it is (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

Proof

technique · direct

1.1F1L1L5

Composition is well defined on bounded-distance classes, by the compatibility lemma.

2.1F1L2L3step 1.1

Associativity is inherited from composition of maps, the class of the identity is neutral, and the class of any chosen quasi-inverse is a two-sided inverse, because the composites are at bounded distance from the identities by definition of quasi-isometry.

3.1F1L1L4step 2.1∎

If $f : X \to Y$ has quasi-inverse $g : Y \to X$ , then $[h] \mapsto [f \circ h \circ g]$ is a homomorphism $Q I (X) \to Q I (Y)$ whose inverse is $[k] \mapsto [g \circ k \circ f]$ ; the compatibility lemma shows both are well defined on bounded-distance classes.

Depends on

Used by

Dependency tree · two levels

23 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)