While reading about the Teichmuller flow, I am reading about the space of lattices $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$.

I could not a find a good way of computing the Teichmuller flow on this quotient space because I have no way of deciding that two elements are in nearby cosets.

$$ X = \left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \mapsto \left(\begin{array}{cc} e^t & 0 \\ 0 & e^{-t} \end{array}\right)X = \left(\begin{array}{ll} a\, e^t & b \, e^t \\ c \, e^{-t} & d \, e^{-t}\end{array}\right)$$

From a numerical point of view, two coefficients are getting very exponentially large and the others are exponentially small and we need row reduction to keep them in a fundamental domain.

How do we keep track of the cosets? Perhaps it is easier to just write the geodesic flow on the unit tangent bundle, but then I still have to keep reverting back to the fundamental domain.

I one had a copy of Arnoux's "Le codage du flot géodésique sur la surface modulaire" but now I can't find it. Many papers talk about come of the math details, but the coding is still rather messy.