# How is a group visually represented?

I don't understand an example on the wkipedia article for Lie groups:

The group given by $$H = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i a \theta } } \end{array} \right) | \theta \in \mathbb { R } \right\} \subset \mathbb { T } ^ { 2 } = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i \phi } } \end{array} \right) | \theta , \phi \in \mathbb { R } \right\}$$ with $$a \in \mathbb { P } = \mathbb { R } \backslash \mathbb { Q }$$ a fixed irrational number,is a subgroup of the torus $$\mathbb { T } ^ { 2 }$$ that is not a Lie group when given the subspace topology.

I don't understand why the group is represented by all these lines in a $$2\pi$$ length square. A portion of the group $$H$$ inside $$\mathbb {T^2}$$ Small neighborhoods of the element $$h \in H$$ are disconnected in the subset topology on $$H$$

Can someone clarify why we have this image ?

Thank you.

• Jan 22, 2019 at 23:02
• It's kind of like rational multiples of an irrational on the number line - they form a kind of "fine dust" which is dense. One could similarly form, for instance, $\Bbb Q\sqrt{2}\times\Bbb R$ within $\Bbb R^2$, and it would be a dense foliage of lines, "parametrized" by the fine dust on the number line. The striations on a torus coming from an irrational slope are the same - indeed if you were to intersect it with any closed circle through the torus, the intersection would be a fine dust on the circle, just like on the number line.
– anon
Jan 23, 2019 at 2:06

I think the picture is meant to illustrate the image of $$\mathbb R$$ under the map $$\theta\mapsto \tau(\theta)=(\theta\bmod1,a\theta\bmod1),$$ or rather, the image of $$[0,A]$$ under $$\tau$$, where $$A\approx 16$$. The image of all of $$\mathbb R$$ under $$\tau$$ would visually fill up the unit square, even though it does not do so set-theoretically. The closure of the image of $$\mathbb R$$ would equal the unit square, however. The point $$h$$ in the picture shows how the set of $$\theta$$ such that $$\tau(\theta)$$ is close to $$h$$ is disconnected: every now and then a big $$\theta$$ makes $$\tau(\theta)$$ come close to $$h$$, and so on.
Why $$\theta \bmod 1$$ and not $$\exp(2\pi i\theta)$$? Because it is hard to put pixels on the page whose row and column indices are complex numbers. You are supposed to imagine the unit square as representing the torus $$\mathbb T^2$$ by thinking the point $$(x,y)$$ on the page represents the point $$(\exp(2\pi i x),\exp(2\pi i y))$$ on the torus.
• Why do we have $A\approx16$ in this image ?If I understand the mod 1 is what separates each parallel line,and it is done because we could not see in the filled out unit square the disconnections,correct ? Jan 23, 2019 at 0:44
• I counted (maybe with some error) 16 line segments crossing the right hand margin, which is where $\approx 16$ comes from. Without the $A$ cutoff the square would be a solid block of ink. Jan 23, 2019 at 0:51