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Properly discontinuous action of groups

  1. Mar 7, 2006 #1
    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
    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]
    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?

  2. jcsd
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