Thread: Discrete Isometries

Let G be a discrete subgroup of Iso(R2). Show that every subgroup of G is also discrete.

Isn't this true simply because it's a subgroup? So the elements of the subgroup are also in G?

Originally Posted by monomoco

Let G be a discrete subgroup of Iso(R2). Show that every subgroup of G is also discrete.

Isn't this true simply because it's a subgroup? So the elements of the subgroup are also in G?

I'm confused, if you are talking about the fact that a subgroup of a discrete subgroup is discrete, this really is just the (more general statement) that a subspace of a discrete space is discrete. In particular, you want to show that each element of is open, but for each you know (via the fact that is discrete) that there exists some open set such that . Clearly though and so is open in . Since was arbitrary the conclusion follows. Make sense?

We're talking about a subgroup of Iso (R2) so by discrete I have the definition:
G< Iso (R2) is discrete there exists an E such that

for all translation t_a in G, a < E
for all rotations in r_o in G, o<E

so the motions cannot be arbitrarily small. I understand what you mean, but I don't know how to apply it to this example.

Originally Posted by monomoco

We're talking about a subgroup of Iso (R2) so by discrete I have the definition:
G< Iso (R2) is discrete there exists an E such that

for all translation t_a in G, a < E
for all rotations in r_o in G, o<E

so the motions cannot be arbitrarily small. I understand what you mean, but I don't know how to apply it to this example.

Oh, well then in that case (the two are equivalent concepts though, which I think you noticed) you must merely note that the same that works for works for . So yes, I think it's as simple as you indicated in the initial post.

I thought I must be missing something. Thanks for your help!