The deck transformation group acts properly discontinuously

Let $𝑝 : ˜ 𝑋 ↠ 𝑋$ be a connected and locally path-connected covering and $Γ = A u t 𝖢 𝗈 𝗏 𝑋 (𝑝)$ be its deck transformation group with its natural action on $˜ 𝑋$ . Then $Γ$ acts on $˜ 𝑋$ properly discontinuously.

Proof

Since $𝖢 𝗈 𝗏 (𝑋, 𝑥 0)$ is thin, $𝛾 1 (˜ 𝑥 0) \neq 𝛾 2 (˜ 𝑥 0)$ for any $𝛾 1, 𝛾 2 \in Γ$ with $𝛾 1 \neq 𝛾 2$ . Let $𝑥 0 = 𝑝 (˜ 𝑥 0)$ , and let $𝑈$ be an evenly covered path-connected open neighbourhood of $𝑥 0$ with $˜ 𝑈$ the sheet over $𝑈$ containing $˜ 𝑥 0$ Since A deck transformation maps sheets to sheets, both $𝛾 1 (˜ 𝑈)$ and $𝛾 2 (˜ 𝑈)$ are sheets over $𝑈$ , and since they each contain a different element of the fibre $𝑝 - 1 {˜ 𝑥 0}$ , they are disjoint. Therefore $Γ$ acts properly discontinuously

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The deck transformation group acts properly discontinuously

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