Asymptotic Winding of the Geodesic Flow on Modular Surfaces and Continuous Fractions

Question

Asymptotic Winding of the Geodesic Flow on Modular Surfaces and Continuous Fractions. Y. Guivarc'h and Y. Le Jan. Page 24.

How to make sure that $K$ and $D$ are normal subgroups of the group $G$ to take the quotients? And, in general, I look for a good reference for such concepts.

Answer 1

Of course these are not normal subgroups, but these are closed subgroups; the quotients are understood as smooth manifolds.