Compactness of the quotient of the plane under isometries. Let's say that $G$ is the group of isometries of $\mathbb{R}^2$ with the usual metric, and $H < G$ is a subgroup generated by one element. Prove that the quotient $\mathbb{R}^2/H$ is either noncompact or not Hausdorff (anyway, I think it is always noncompact...)
The thing is, I can visualize this when $H$ leaves the origin fixed. But when $H$ is made of compositions of translations and rotations it's really bad, I don't see how the quotient turns out to be...
Any help would be appreciated!

A single isometry, with the exception of the identity, can be classified as one of the following:


*

*translation

*rotation

*reflection

*glide reflection


A composition of a translation and a rotation is again a rotation, just around a different center. For discrete groups, your quotient space will be compact if and only if $H$ contains two linearily independent translations. This would be the case of wallpaper groups. But since your $H$ is generated by a single transformation, that shouldn't worry you. So let's go through the alternatives.
Translation
A group generated by a single translation will be the frieze group p1. Its quotient space is an infinite strip, with edges identified, but since it extends towards infinity it is not compact.
Rotation
A rotation by a rational fraction of $2\pi$ will give you a cyclic group as symmetry group, and a glued up pizza slice (also known as a cone) as quotient space. Again non-compact.
For an irrational fraction of $2\pi$ your pizza slice becomes arbitrarily thin (see comments below for more thoughts on that), but there are still infinite sequences along a ray in radial direction, so this is again non-compact.
Reflection
This is an involution, the quotient space is a half plane. Not compact.
Glide reflection
The symmetry group is frieze group p11g, and you can again choose a strip of the plane as fundamental domain. Same argument as for translations tells you the quotient space is again non-compact.
Identity
The quotient space of the identity transformation is the original space, very clearly non-compact.
