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I'm not sure if the following definitions of "properly discontinuous" (=:ED) are equivalent. I found them in two different books:

1) G: group

M: top. manifold

G is ED [tex]\Leftrightarrow[/tex] for all compact [tex] K \in M[/tex] there only finitely [tex]g_i \in G[/tex] exist with [tex]g_i(K) \cap K \neq \emptyset[/tex]

2) G:discrete group (finite or countable infinite with discrete

top.)

G acts continuously on M

G is ED [tex]\Leftrightarrow[/tex]

i) every [tex]p \in M[/tex] has a neighbourhood U with [tex](g*U) \cap U

=\emptyset[/tex] only for all but finitely many [tex]g \in G[/tex]

and

ii) If [tex]p, p' \in M[/tex] are not in the same G-orbit, there exist neighborhoods U of p and U' of p' such that [tex](g*U) \cap U^{'} = \emptyset \ \forall g \in G[/tex]

could you help me?

Paul

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# Properly discontinuous action of groups

Can you offer guidance or do you also need help?

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