# Properly discontinuous action of groups

1. Mar 7, 2006

### F.Prefect

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