LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-26

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Orbit maps of isometric actions are coarse Lipschitz

Statement

Let a finitely generated group $G$ with finite generating set $S$ act isometrically on a metric space $X$ , and fix $x_{0} \in X$ . Then the orbit map $ϕ_{x_{0}} : (G, d_{S}) ⟶ X, ϕ_{x_{0}} (g) : = g \cdot x_{0},$ is coarse Lipschitz. In fact, if $M : = \max ({0} \cup {d_{X} (x_{0}, s \cdot x_{0}) : s \in S \cup S^{- 1}}),$ then $d_{X} (g \cdot x_{0}, h \cdot x_{0}) \leq M d_{S} (g, h) for all g, h \in G .$

Facts & Assumptions

Given: A finite generating set $S$ of $G$ , an isometric action of $G$ on a metric space $X$ , and a point $x_{0} \in X$ .

[L1]

A group is finitely generated when some finite subset generates it (Finitely generated groups).

[L2]

An isometric action satisfies $d (g \cdot x, g \cdot y) = d (x, y)$ for all $g \in G$ and $x, y \in X$ (Isometric, proper, and cobounded actions on metric spaces).

[L3]

The word metric is $d_{S} (g, h) = ∣ g^{- 1} h ∣_{S}$ (The word metric of a group with respect to a generating set), and $∣ u ∣_{S}$ is the least length of an expression of $u$ as a product of elements of $S \cup S^{- 1}$ (Word length of a group element with respect to a generating set).

[L4]

A map is coarse Lipschitz when its output distances are bounded by $A$ times the input distance plus an additive constant $B$ , for some reals $A, B \geq 0$ (Coarse Lipschitz maps and quasi-isometric embeddings).

Proof

technique · direct

1.1L1choose

Because $S \cup S^{- 1}$ is finite by [L1], adjoining $0$ gives a nonempty finite set of real numbers, so the maximum $M$ exists.

2.1L2L3step 1.1algebra

Let $u : = g^{- 1} h$ , and write $u = s_{1} \dots s_{n}$ with $n = ∣ u ∣_{S} = d_{S} (g, h)$ and each $s_{i} \in S \cup S^{- 1}$ by [L3]. Repeated use of the triangle inequality gives $d_{X} (x_{0}, u \cdot x_{0}) \leq \sum_{i = 1}^{n} d_{X} (x_{0}, s_{i} \cdot x_{0}) \leq n M$ . Applying the isometry $g$ and [L2] yields $d_{X} (g \cdot x_{0}, h \cdot x_{0}) = d_{X} (x_{0}, u \cdot x_{0}) \leq M d_{S} (g, h)$ .

3.1L4step 2.1∎

The displayed estimate is a coarse-Lipschitz bound with multiplicative constant $M$ and additive constant $0$ , so the orbit map is coarse Lipschitz by [L4].

Depends on

Used by

The Svarc-Milnor lemma Theorem

Dependency tree · two levels

15 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. 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)