Let $T$ be a torus (oriented closed surface of genus 1), $p\in T$, and $T^* := T - \{p\}$.

Let $\mu$ denote a flat structure on $T$. This can be obtained for example by choosing a uniformization $p_f: T\cong\mathbb{R}^2/\Lambda$ for some lattice $\Lambda$ and descending the standard flat structure on $\mathbb{R}^2$ down to $T$.

The flat structure on $T$ restricts to one on $T^*$. Moreover, if we fix an identification $\mathbb{C}\cong\mathbb{R}^2$, this flat structure gives a complex structure on $T^*$. This complex structure yields a holomorphic universal covering $p_h : \mathbb{H}\rightarrow T^*$, such that the deck transformation group consist of hyperbolic isometries of $\mathbb{H}$, and hence this complex structure determines a hyperbolic structure on $T^*$.

By a paper of Gutkin and Judge (Affine mappings of translation surfaces), it seems to follow from their Theorem 6.5 that the number of (simple) closed geodesics on a flat (punctured) torus (up to translation) of length bounded by $L$ is quadratic in $L$. If the torus is the unit square torus, the leading term turns out to be $\frac{3}{\pi}L^2$.

On the other hand, by a result of McShane and Rivin (A norm on homology of surfaces and counting simple geodesics, as pointed out to me in a previous question), there is a natural bijection between the simple closed geodesics in the flat torus up to translation with those in the corresponding hyperbolic torus (both sets are bijective onto the set of primitive homology classes under the natural map), and that the number of hyperbolic simple closed geodesics of length bounded by L is also quadratic in L.

This suggests the flat geodesic representatives of primitive homology classes might have length which is universally proportional to that of the hyperbolic geodesic representative. Could this be true? If not, what is known about the relationship between these lengths under the natural bijection described above?

References would be appreciated!


1 Answer 1


Yes, this is true. It is shown (either in the paper you cite or the other McShane-Rivin paper) that the length of a simple closed geodesic is quasi-the-same as the combinatorial length ($m+n$) (this is easy, because a simple geodesic stays away from the cusp), and that, in turn, is easily seen to be quasi-the-same as the Euclidean length.


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