  We extend the Series' connection between the modular surface
$\mathcal{M}=\operatorname{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$, cutting
sequences, and regular continued fractions to the slow converging Lehner and
Farey continued fractions with digits $(1,+1)$ and $(2,-1)$ in the notation
used for the Lehner continued fractions. We also introduce an alternative
insertion and singularization algorithm for Farey expansions and other
non-semiregular continued fractions, and an alternative dual expansion to the
Farey expansions so that $\frac{dxdy}{(1+xy)^2}$ is invariant under the natural
extension map.
