Let $G$ be a Hausdorff topological group. We say that a subgroup $S$ of $G$ is discrete if and only if the subspace topology (from $G$) on $S$ is discrete.

Note that isometry group of euclidean space $\mathbb{R}^n$ is displayed by $E(n)$(or affine isometries on $\mathbb{R}^n , \ O(n)\subset E(n)$) and $Sx:=\left\{s(x);\ \forall s\in S\subseteq E(n)\right\}$ is $S$-orbit of any $x\in\mathbb{R}^n$ where S is any subgroup of E(n).

For discrete subgroups S of isometry group of euclidean space $E(n)$ an equivalent condition is: intersection of the $S$-orbit of any $x\in\mathbb{R}^n$ has finite intersection with any compact set of $\mathbb{R}^n$.

Why is there a such the equivalent condition? How can I prove it?

  • $\begingroup$ Affine isometries or $O(n)$? Anyway, recall how you definie the topology on $G$ $\endgroup$ – Hagen von Eitzen Nov 25 '13 at 17:29

The definition means that $S$ is discrete if for each point $x\in \mathbb{R}^n$ the family $Sx$ is locally finite, i.e., for each point there is a neighbourhood intersecting only finitely many subsets of this family. This is equivalent to the fact each compact set intersects only finitely many subsets of this family.

  • $\begingroup$ To add to this solution: The only properties you need are that the group action is isometric and that closed balls are compact. $\endgroup$ – Moishe Kohan Nov 25 '13 at 20:30
  • $\begingroup$ What's your mean from the family $Sx$? Is your mean $\mathcal{F}=\{\text{singleton subsets of} \ Sx\}$ for any $x\in \mathbb{R}^n$ that fixed? Why is this family locally finite? I do not figure out The last equivalence. Why are these property only for group action of isometries on $\mathbb{R}^n$? Please describe them more than your posts. $\endgroup$ – bigli Nov 26 '13 at 5:32
  • $\begingroup$ I mean $Sx=\{s.x \mid s \in S\}$. The word "family" is used in Vinberg's book on discrete groups. Your definition of discrete is just that each Sx is locally finite, see en.wikipedia.org/wiki/Subspace_topology. $\endgroup$ – Dietrich Burde Nov 26 '13 at 8:36
  • $\begingroup$ Dear Dietrich Burde, My $\mathcal{F}$ is the same with your mean from $Sx=\{s.x\ | s\in S\}$. Why is this family($\mathcal{F}$) locally finite? I do not figure out The last your equivalence in your answer. What's Vinberg's book? What's connection the Wikipedia webpage(in the last your comment) with my problem? $\endgroup$ – bigli Nov 27 '13 at 3:37
  • $\begingroup$ Dear bigli, Vinberg's book: Geometry II. Encyclopedia of Mathematical Sciences, Springer 1991. The definition of the topology answers your question, see the remark of Hagen von Eitzen. But you can find all details in Vinberg's book. $\endgroup$ – Dietrich Burde Nov 27 '13 at 8:57

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