bibkey: kapovich2023properactions authors: M. Kapovich year: 2023 title: A note on properly discontinuous actions doi: 10.1007/s40863-023-00353-z url: https://www.math.ucdavis.edu/~kapovich/EPR/prop-disc.pdf claim: Lemma 21(2) proves properness of an isometric orbit quotient of a proper metric space by projection of closed balls; the preceding discussion identifies proper and metrically proper discrete actions in a proper ambient space. strata_touched:

• D5/S3/Geometry/IsometricOrbitMetric license: citation-only triage: anchor

Compatible isometric orbit metrics and quotient properness

Source and scope

Kapovich, A note on properly discontinuous actions, published in 2023. The author’s revised manuscript at the URL above is dated 18 June 2024. The locators here refer to that manuscript: page 9, the quotient metric formula (20) and Lemma 21(2); page 10, its closed-ball projection proof. The author’s introduction reports a correction to Section 7 concerning fundamental domains. This note uses the Section 6 quotient metric argument. The journal also lists a correction, DOI 10.1007/s40863-024-00457-0; no claim about the contents of that separate correction is made here.

For an isometric discrete action on a proper metric space, the manuscript identifies properness of the action with metric properness. It then gives a quotient metric inducing the quotient topology and proves that the quotient is proper: each closed quotient ball is the projection of the closed ambient ball about a representative. Neither freeness nor geodesicity is required for Lemma 21(2). No cocompactness hypothesis is needed.

Representation convention and formal construction

Let G be a group, X a metric space, and rho a homomorphism from G to the self-isometries of X. Write pi:X -> X/rho for the orbit projection. The formal construction uses the infimum distance to an orbit rather than asserting a minimum without ambient properness. It proves metric and quotient-topology compatibility for compact-set proper discontinuity even when X is not proper. The properness conclusion additionally assumes that X is proper. The general nonproper-ambient clause is not attributed to Kapovich’s metrically proper-action statement alone.

These are classical quotient-metric constructions and their formal bridges; this note makes no mathematical novelty or rigidity claim. The representation quotient convention agrees with the standard orbit quotient. Mathlib’s proper-action Hausdorff theorem, closed-set zero-distance criterion and nearest-point attainment supply the corresponding ingredients.

Infimum metric and its proof

For the representation orbit relation, put (O_y={\rho(g)y:g\in G}) and define

[ d_\rho([x],[y])=\operatorname{infDist}(x,O_y) =\inf_{g\in G}d_X(x,\rho(g)y). ]

The orbit contains (y), so this infimum is finite and nonnegative. Distance to a nonempty set, its invariance under an isometry, and attainment on a nonempty closed subset of a proper metric space are the standard point-to-set distance results.

For the representation quotient, the displayed function is independent of representatives. If the representation is properly discontinuous in the compact-set sense, this function is a metric inducing the existing quotient topology. If in addition (X) is proper, then ((X/\rho,d_\rho)) is proper. Freeness and compactness of the quotient are not hypotheses.

Proof. Replacing (x) by (\rho(a)x) and (y) by (\rho(b)y) reindexes the orbit: (\rho(a)O_y=O_{\rho(b)y}). Isometry invariance of point-to-set distance proves independence. Inversion in (G) and distance symmetry give symmetry of (d_\rho). For all (g,h\in G),

[ d_\rho([x],[z])\leq d_X(x,\rho(gh)z) \leq d_X(x,\rho(g)y)+d_X(y,\rho(h)z). ]

Taking the two infima proves the triangle inequality. The distance from an orbit to itself is zero. Proper discontinuity on a metric space gives a proper action when the group is given the discrete topology, and a proper action has a Hausdorff orbit quotient. These standard proper-action results apply through the standard orbit quotient comparison. Thus each fiber (O_y=\pi^{-1}{[y]}) is closed. The standard zero-distance criterion for a nonempty closed set gives (d_\rho([x],[y])=0) exactly when (x\in O_y), proving separation.

If (U) is quotient-open and ([x]\in U), some ambient ball about (x) lies in (\pi^{-1}U). Any class sufficiently close to ([x]) has a translate of a representative in that ball, so belongs to (U). Conversely, the inequality (d_\rho([x],[y])\leq d_X(x,y)) shows that the preimage of an orbit-distance neighborhood contains an ambient ball. These two implications identify the metric topology with the existing quotient topology.

Suppose now that (X) is proper. Distance from (x) to the closed nonempty orbit (O_y) is attained at a point (z\in O_y). If (d_\rho([x],[y])\leq r), then (d_X(x,z)\leq r) and ([z]=[y]), which gives one inclusion in the ball identity. The opposite inclusion follows from (d_\rho([x],[z])\leq d_X(x,z)). The ambient closed ball is compact and (\pi) is continuous in the agreed topology, so the quotient closed ball is compact. This bridge concerns the representation quotient; it imposes no finite-volume or smooth-curvature conclusion.

Verified locator

• DOI: https://doi.org/10.1007/s40863-023-00353-z
• Author manuscript: https://www.math.ucdavis.edu/~kapovich/EPR/prop-disc.pdf
• Verified scope: revised manuscript dated 18 June 2024, page 9, formula (20) and Lemma 21(2), with the closed-ball projection proof on page 10.