Closure of a one-parameter subgroup

[eok20]

I'm reading about a theorem that has as an assumption that the closure of some one-parameter subgroup is a torus. Could someone provide an example of a case where the closure of a one-parameter subgroup is of dimension greater than 1?

Thanks.

 

[lavinia]

I'm reading about a theorem that has as an assumption that the closure of some one-parameter subgroup is a torus. Could someone provide an example of a case where the closure of a one-parameter subgroup is of dimension greater than 1?

Thanks.

The usual example pictures the torus as a square with opposite edges identified cylindrically.

On the square pick any starting point and draw a straight line at some angle to one of the edges. When the line hits an edge of the square continue the line starting on the opposite edge. Keep doing this. This generates a 1 parameter subgroup of the torus viewed as the group,

$$S^1 \times S^1$$

you should prove

- the line describes a closed curve on the torus iff the angle is a rational number

- if the angle is an irrational number the line's image on the torus is dense i.e. it comes arbitrarily close to any point on the torus.

Since the image is dense, its closure is the whole torus.

If you do not require the curve to be a subgroup then there are examples that actually completely fill the torus or for that matter any compact region of a manifold.