Group action

Properly discontinuous group action

A group 𝐺 acting on a topological space 𝑋 is called properly discontinuous¹ iff every 𝑥 ∈𝑋 has a neighbourhood 𝑈 such that for every 𝛾1,𝛾2 ∈𝐺 with 𝛾1 ≠𝛾2, 𝛾1𝑈 ∩𝛾2𝑈 =∅. topology

Properties

1. A properly discontinuous group action is necessarily free.
2. If 𝐺 is also topological group and acts continuously, then the orbit map 𝐺 →𝐺𝑥 is a homeomorphism of discrete topological spaces.

Proof of properties 1–2

Let 𝐺 be a topological group acting on 𝑋 continuously and properly discontinuously.

Assume that 𝐺 does not act freely, i.e. there exist 𝛾1,𝛾2 ∈𝐺 with 𝛾1 ≠𝛾2 such that 𝛾1𝑥 =𝛾2𝑥 for some 𝑥. Then for any neighbourhood 𝑈 of 𝑥, 𝛾1𝑥 =𝛾2𝑥 ∈𝛾1(𝑈) ∩𝛾2(𝑈), violating proper discontinuity. Thus 𝐺 acts freely.

Now consider the orbit of a point 𝑥 with its subspace topology and the corresponding orbit map 𝑜𝑥 :𝐺 →𝐺𝑥 :𝛾 ↦𝛾𝑥.

Assume there exists 𝛾1𝑥 ∈𝐺𝑥 with {𝛾1𝑥} not open in 𝐺𝑥. Let 𝑈 be an open neighbourhood of 𝑥 in 𝑋. Since 𝛾1 is a homeomorphism, 𝛾1𝑈 is open in 𝑋, and thus 𝛾1𝑈 ∩𝐺𝑥 is open in 𝐺𝑥, so at least one distinct point 𝛾2𝑥 is contained in 𝛾1𝑈. Then 𝛾2𝑥 ∈𝛾1𝑈 ∩𝛾2𝑈, violating proper discontinuity. Therefore 𝐺𝑥 must be discrete.

Now clearly the orbit map 𝑜𝑥 is continuous and bijective (injectivity by freeness, surjectivity by construction). Thus every singleton {𝛾1} in 𝐺 is the preïmage of a singleton {𝛾1𝑥} in 𝐺𝑥 and is therefore open. Therefore 𝐺 is discrete, and 𝑜𝑥 is a homeomorphism, since the inverse 𝑜−1𝑥 is continuous as a map between discrete spaces.

3. Orbit space of a properly discontinuous group action 𝑋/𝐺 covers 𝑋.

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Footnotes

1. German eigentlich diskontinuierlich ↩