If G is an infinite planar group (it means that it has a generating subset C such that Cay (S, C) is a planar graph) and H is a normal subgroup of it, I would be very grateful if somebody helps me and tell me "is G/H a planar group?"



quotient of planar groups
The dihedral groups can be viewed as the set of all functions of the form $x\mapsto\pm x+c$ acting either on $\mathbb{Z}$ or on $\mathbb{Z}/n\mathbb{Z}$. The images of the infinite dihedral group are either finite dihedral groups. Taking as generators $x\mapsto x+1$ and $x\mapsto -x$ you get for the dihedral group of order $2n$ a planar Cayley graph consisting of two circles of length $n$.


Hence the infinite dihedral group is an example of an infinite non-abelian group, which has only planar images. I tried to find a more interesting example, but failed.


