F.Prefect
Mar7-06, 12:18 PM
Hello,
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 \Leftrightarrow for all compact K \in M there only finitely g_i \in G exist with g_i(K) \cap K \neq \emptyset
2) G:discrete group (finite or countable infinite with discrete
top.)
G acts continuously on M
G is ED \Leftrightarrow
i) every p \in M has a neighbourhood U with (g*U) \cap U
=\emptyset only for all but finitely many g \in G
and
ii) If p, p' \in M are not in the same G-orbit, there exist neighborhoods U of p and U' of p' such that (g*U) \cap U^{'} = \emptyset \ \forall g \in G
could you help me?
Paul
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 \Leftrightarrow for all compact K \in M there only finitely g_i \in G exist with g_i(K) \cap K \neq \emptyset
2) G:discrete group (finite or countable infinite with discrete
top.)
G acts continuously on M
G is ED \Leftrightarrow
i) every p \in M has a neighbourhood U with (g*U) \cap U
=\emptyset only for all but finitely many g \in G
and
ii) If p, p' \in M are not in the same G-orbit, there exist neighborhoods U of p and U' of p' such that (g*U) \cap U^{'} = \emptyset \ \forall g \in G
could you help me?
Paul