I was listening to this lecture: and in it, sometime around the 30:00 to 40:00 minute mark, he implies that the torus' sturcture built up from the orbits of the group under addition on the real plane is the same idea as the cylinder's structure being built up from the orbits of the group under addition on the real line. This makes sense to me, except for the fact that it seems to me that in the case of the cylinder, it shouldn't have any height in order to be analogous to the torus. Moving around the circumference of the cylinder is equated with cycling through the different orbits, similar to moving along either circle in the torus. However there is nothing in the torus analogous to the moving along the height of the cylinder, and so it seems to me that a better analogy would use a circle instead of a cylinder. Is that the case, or am I missing something?

The cylinder is the space of orbits of the action of the integers on the plane. The action can be taken to be horizontal translation of the plane by an integer amount. ##n⋅(x,y) = (x+n,y)##. The orbit space of the action of the integers on the real line ##n⋅x=x+n## is a circle.

For the torus the group is different. It is the group ##Z×Z## the group of pairs of integers ##(n,m)## under coordinatewise addition. Its action on the plane is ##(n,m)⋅(x,y) = (n+x,m+y)##.