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.

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I one had a copy of Arnoux's "[Le codage du flot géodésique sur la surface modulaire](http://retro.seals.ch/digbib/view?pid=ensmat-001:1994:40::185)" but now I can't find it.  Many papers talk about some of the math details, but the coding is still rather messy.

In response to recent questions about the meaning of the word **compute** I have found an entry from the dictionary as well as [9,380](https://mathoverflow.net/search?q=compute) instances of the word on MathOverflow and [38,883](https://math.stackexchange.com/search?q=compute) instances on Math.SE

![enter image description here][1]


  [1]: https://i.sstatic.net/X4BOV.png