Deck transformation

The deck transformation group acts properly discontinuously

Let be a connected and locally path-connected covering and be its deck transformation group with its natural action on . Then acts on properly discontinuously.

Proof

Since is thin, for any with . Let , and let be an evenly covered path-connected open neighbourhood of with the sheet over containing Since A deck transformation maps sheets to sheets, both and are sheets over , and since they each contain a different element of the fibre , they are disjoint. Therefore acts properly discontinuously

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