There exists a a metric space such that its group of isometries is isomorphic to $\mathbb{Z}$.

Question

My question is as follows: I have to choose true or false. This is a question from a Graduate school admission test.

There exists a metric space such that its group of isometries is isomorphic to $\mathbb{Z}$.

Is it true?

Does this answer your question? $\mathbb{Z}$ is the symmetry group of what? — Didier, Dec 23 '21 at 09:35

orangeskid · Accepted Answer · 2021-12-23T09:31:39.113

5

HINT:

The solution of @MathematicsStudent1122 is quite salvageable, just consider the space $$\mathbb{Z} \cup (\mathbb{Z}+ \frac{1}{3})\cup (\mathbb{Z}+\frac{1}{2})$$

edited Dec 23 '21 at 09:31

answered Dec 23 '21 at 08:16

orangeskid

56,630

@MathematicsStudent1122: it is basically your idea + a symmetry break – orangeskid Dec 23 '21 at 08:18
1

Thanks! This is great – MathematicsStudent1122 Dec 23 '21 at 08:18
@MathematicsStudent1122: It was still not right, needed another translate to break the symmetry, now I think it's OK. It's basically one of the frieze patterns. – orangeskid Dec 23 '21 at 09:33
2

can someone please elaborate a little further – ogirkar Feb 19 '24 at 14:01

There exists a a metric space such that its group of isometries is isomorphic to $\mathbb{Z}$.

1 Answers1