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Let $G$ be a discrete subgroup of $Iso(\mathbb R^2)$. Show that every subgroup of $G$ is discrete.

Is it enough to say that since any element of a subgroup of $G$ is also in $G$ it satisfies the condition that it's greater than or equal to some $ \epsilon > 0$ then that subgroup must also be discrete?

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1 Answer 1

Hum, I think the question needs another edit (it is formulated correctly but I can't make sense of the suggested answer).

I think the argument you're looking for is : if $g \in H$ is an element of a subgroup $H$ then there is $\epsilon >0$ such that $B(g, \epsilon)$ does not meet any element of $G$ (since $G$ is discrete), hence of $H$. Therefore $H$ is discrete.

Actually the exact same reasonning shows that any part of a discrete topological space is discrete.

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Can you explain what B(g,ϵ) means? –  user19961 Nov 23 '11 at 0:39
    
it means : the set of the $h \in G$ such that the distance between $g$ and $h$ is less thant $\epsilon$. so it depends on the choice of the distance on Iso(R²). –  Glougloubarbaki Nov 23 '11 at 6:33

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