# Continued Fractions and the Modular Surface

Luis B. Castro
(Institut for Matematik)
Foredrag for studerende
Fredag, 2 oktober, 2015, at 15:15-16:00, in Aud. D4 (1531-219)
Abstrakt:
A continued fraction is a formal expression
$\left[ a_0; a_1, a_2, \dots \right] = a_0 + \cfrac{ 1 }{ a_1 + \cfrac{ 1 }{ a_2 + \ddots } },$
where $a_0 \in \mathbb{Z}$ and $a_n \in \mathbb{N}$ for all $n \ge 1$.
There is a one to one correspondence with continued fractions and real numbers
But there is also a correspondence between continued fractions and endpoints of oriented geodesics on the modular surface $M = \mathbb{H} / PSL(2,\mathbb{Z})$ where $\mathbb{H}$ is the hyperbolic plane.

If we look at $\mathbb{H}$ we can tesselate it by the Farey tesselation $\mathbb{F}$ which consists of triangles.
If we then have an oriented geodesic $\overline{ \gamma }$ on $M$ we can lift it to a geodesic $\gamma$ on $\mathbb{H}$.
We can define a cutting sequence $\dots L^{ n_{ -1 } } R^{ n_0} L^{ n_1 } \dots$.
This cutting sequence tells us how $\gamma$ cuts the triangles of $\mathbb{ F }$ but it turns out that it also inculdes information about the left and right endpoints ($\gamma_{ - \infty }$ and $\gamma_\infty$ respectively).
Namely
$\gamma_\infty = \left[ n_1; n_2, \dots \right], \quad \frac{ - 1 }{ \gamma_{ -\infty } } = \left[ n_0; n_{ -1 }, \dots \right].$
After having established this I will use this to prove that the continued fraction representation of a real number $\alpha$ becomes periodic if and only if $\alpha$ is a quadractic irrational i.e. $\alpha$ is root of a 2nd degree polynomial with integer coefficients.