Properly discontinuous action of groups

[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

 

[pat_connell]

Yes, the two definitions of properly discontinuous are equivalent. The first definition states that a group G is properly discontinuous if for any compact set K in a topological manifold M, there are only finitely many elements g_i of G such that g_i(K) intersects K. The second definition states that a discrete group G is properly discontinuous if every point p in M has a neighborhood U such that g*U intersects U for only finitely many elements g of G, and if two points p and p' in M are not in the same G-orbit, then there exist neighborhoods U of p and U' of p' such that g*U intersects U' for all g in G. The two definitions are equivalent because you can use the second definition to prove the first. Given a compact set K in a topological manifold M, you can consider the set of all neighborhoods of K. This set is finite, so the second definition implies that there are only finitely many elements g_i of G such that g_i(K) intersects K. Thus, the two definitions are equivalent.

 

[tacman]

Hello Paul,

Thank you for reaching out. Yes, these two definitions are equivalent. Let's break them down to see why:

1) In the first definition, we have a group G acting on a topological manifold M. This means that every element g in G is a homeomorphism on M, preserving the topological structure. Now, for G to be properly discontinuous, this means that for any compact set K in M, there are only finitely many elements g_i in G such that the image of K under g_i intersects with K. In other words, there are only finitely many elements in G that can "move" K in such a way that it intersects with itself.

2) In the second definition, we have a discrete group G acting continuously on M. This means that the action of G on M is continuous, meaning that for any point p in M, there is a neighborhood U of p such that for all g in G, the image of U under g is also a neighborhood of p. Now, for G to be properly discontinuous, this means that for any point p in M, there is a neighborhood U of p such that for all but finitely many elements g in G, the image of U under g does not intersect with U. This is essentially the same as the first definition, but instead of considering a compact set K, we are considering a neighborhood U of a point p.

So, both definitions are essentially saying the same thing: for a group to be properly discontinuous, there are only finitely many elements that can "move" a set or a point in such a way that it intersects with itself or another set/point. I hope this helps clarify things for you.