How to find representatives of $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$

Question

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 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 instances of the word on MathOverflow and 38,883 instances on Math.SE

Answer 1

I am not sure what you mean by "computing" the Teichmuller flow, but the standard way to visualize the flow is through the Poincare upper-half plane model.

Recall that $PSL_2({\bf Z})$ acts through homographies on the Poincare half-plane ${\bf H}=\{z=x+iy \in {\bf C} \mid Im(z)>0\}$ and on its unit tangent bundle $T^1{\bf H}$. This bundle is identified to $PSL_2({\bf R})$ through the isomorphism $$ \Psi\Bigl( {\small \pmatrix{1 & x \cr 0 & 1 \cr}} {\small \pmatrix{\sqrt{y} & 0 \cr 0 & 1/\sqrt{y} \cr}} {\small \pmatrix{\cos({\theta\over 2}) & \sin({\theta\over 2}) \cr -\sin({\theta\over 2}) & \cos({\theta \over 2}) \cr}}\Bigr) = (x,y,\theta) $$ where $\theta$ is the angle between the vector and the vertical direction.

The quotient of $T^1{\bf H}$ by the action of $PSL_2({\bf Z})$ is isomorphic to $SL_2({\bf R})/SL_2({\bf Z})$. It is customary to represent the quotient space ${\bf H}/PSL_2({\bf Z})$ through the Dirichlet fundamental domain of $2i$, which is given by $$ D = \{ z\in {\bf H} \mid |z| \geq 1, \ |Re(z)| \leq 1/2\}. $$

Your Teichmuller flow is the geodesic flow associated to the -1 curvature metric on $\bf H$ and the projection of its trajectories on $D$ are circles orthogonal to the boundary of $\bf H$. So what happens when such circle crosses the boundary of $D$? If it crosses one of the two vertical lines, then you translate it with $z\mapsto z \pm 1$ to bring it back in $D$. If it crosses the piece of unit circle, you apply $z\mapsto -1/z$ and then you translate it back to the central strip. Iterate until you reach $D$, don't forget to keep track of the angle. Translate it back to its matrix form and this gives you the representative you are looking for.

Geometrically, the trajectory of the flow is made of pieces of circles in $D$ that start to fill $D$ (or $T^1D$) densely for almost all initial starting vectors.

If you look at the action of $z\mapsto z+1$ and $z\mapsto -1/z$ on the end points of the circle that lie on the boundary of $\bf H$, you can see that this is just the standard algorithm that computes the continuous fraction decomposition of the (endpoints of the trajectory associated to the) initial vector. This is the standard coding of the geodesic flow on the modular surface, described in a famous paper of Caroline Series ("The modular surface and continued fractions", jlms, 1985).