ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-26

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Scaling maps embed the multiplicative group of nonzero reals into the quasi-isometry group of $Z$

Example

Scaling maps embed the multiplicative group of nonzero reals into the quasi-isometry group of $Z$ .

Facts & Assumptions

Given: The objects and hypotheses in the Example.

[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]

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).

[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]

It is written $⌊ x ⌋$ and called the integer part, or floor, of $x$ . (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ).

[L4]

Define $d_{R} : R \times R \to R$ by $d_{R} (x, y) : = ∣ x - y ∣$ . (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded).

[L5]

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

Verification

technique · direct

1.1F1L2L3L4

For $α \neq 0$ let $q_{α} (n) = ⌊ α n ⌋$ . The estimate $∣ ⌊ u ⌋ - ⌊ v ⌋ ∣ \leq ∣ u - v ∣ + 1$ shows $q_{α}$ is coarse Lipschitz, and $q_{1 / α}$ is a quasi-inverse because $∣ q_{α} (q_{1 / α} (n)) - n ∣ < ∣ α ∣ + 1 and ∣ q_{1 / α} (q_{α} (n)) - n ∣ < ∣ 1 / α ∣ + 1$ for every integer $n$ . So $q_{α}$ is a quasi-isometry of $Z$ .

2.1L3step 1.1

Composing the maps for $α$ and $β$ agrees with the map for $α β$ up to an error of at most $∣ α ∣ + 1$ , so the assignment $α \mapsto [q_{α}]$ is a homomorphism on classes.

3.1F1L1L5step 2.1∎

For $α \neq β$ the difference $∣ ⌊ α n ⌋ - ⌊ β n ⌋ ∣$ is unbounded, so the homomorphism is injective.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

36 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)

Scaling maps embed the multiplicative group of nonzero reals into the quasi-isometry group of ZZ