Connected fundamental domains for congruence subgroups

We produce canonical sets of right coset representatives for the congruence subgroups ${\mathrm{\Gamma }}_0(N)$, ${\mathrm{\Gamma }}_1(N)$ and $\mathrm{\Gamma }(N)$, and prove that the corresponding fundamental domains are connected. Key to our construction is a study of the projective line $P^1(\mathbb{Z}\mathrm{/}N\mathbb{Z})$ using a function $M:\mathbb{Z}\mathrm{/}N\mathbb{Z}\to {\mathbb{Z}}_{\ge 0}$, representing multiplicities. We further study this function and show that it is simply one less than another much more computable function $W:\mathbb{Z}\mathrm{/}N\mathbb{Z}\to \mathbb{N}$, of possible independent interest. We present some examples and pictures at the end.