LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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Metric properness agrees with proper discontinuity on proper discrete metric spaces

Statement

Let $G$ act isometrically on a proper discrete metric space $X$ . Then the metric-properness condition of Isometric, proper, and cobounded actions on metric spaces is equivalent to the usual proper-discontinuity condition that for every finite subset $F \subseteq X$ , the set ${g \in G : (g \cdot F) \cap F \neq \emptyset}$ is finite.

Equivalently, it is enough to require that for every $x \in X$ and every $R \geq 0$ , the set ${g \in G : d (x, g \cdot x) \leq R}$ be finite.

Facts & Assumptions

Given: An isometric action of $G$ on a proper discrete metric space $X$ .

[L1]

The action is proper when, for every bounded subsets $B, C \subseteq X$ , the transporter set ${g \in G : (g \cdot B) \cap C \neq \emptyset}$ is finite (Isometric, proper, and cobounded actions on metric spaces).

[L2]

In a proper discrete metric space, bounded subsets are finite. [given]

Proof

technique · direct

1.1

If the action is proper in the metric sense, then [L2] turns every finite set into a bounded set. So for every finite $F \subseteq X$ , the set ${g \in G : (g \cdot F) \cap F \neq \emptyset}$ is finite by [L1].

L1L2

1.2

Conversely, suppose the finite-set condition holds. Let $B, C \subseteq X$ be bounded. By [L2], the union $F : = B \cup C$ is finite. If $(g \cdot B) \cap C \neq \emptyset$ , then certainly $(g \cdot F) \cap F \neq \emptyset$ , so the transporter of $B$ into $C$ is contained in the finite set supplied for $F$ . Hence the action is proper in the metric sense.

L1L2

2.1

The finite-set condition implies the pointwise bound by taking $F : = \overset{ˉ}{B} (x, R)$ ; and the pointwise bound implies the finite-set condition because a finite set lies in some ball $\overset{ˉ}{B} (x, R)$ . Thus all three formulations are equivalent.

step 1.1step 1.2algebra∎

Depends on

Used by

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Dependency tree · two levels

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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)