Euler characteristics of $SL(2,\mathbb{Z})$-orbit graphs of origamis

Abstract

The $SL(2,\mathbb{Z})$-orbits of primitive $n$-squared origamis can be represented by finite four-regular graphs. It is a conjecture of McMullen that the orbit graphs of such origamis in the stratum $\mathcal{H}(2)$ form an expander family. We provide indirect evidence for this conjecture by proving that the absolute values of the Euler characteristics of the graphs in this family go to infinity with the number of squares $n$. This generalises previous work of the authors, in which we established eventual non-planarity for this family, and provides the strongest indirect evidence to date for McMullen's conjecture. We also prove that the same phenomenon holds for primitive origamis in the Prym loci of $\mathcal{H}(4)$ and $\mathcal{H}(6)$. Assuming conjectures of Zmiaikou and Delecroix--Leli\`evre, we establish the same in $\mathcal{H}(1,1)$ and for two families of non-Prym origamis in $\mathcal{H}(4)$. Finally, assuming a stronger conjecture concerning orbit growth in low-complexity strata, we establish that for any family of $SL(2,\mathbb{Z})$-orbit graphs of primitive origamis in a stratum with one or two conical singularities, the absolute values of the Euler characteristics of the graphs go to infinity with the number of squares along a density one subsequence. By relating the genus of the elliptic generator orbit graphs to the genus of the associated arithmetic Teichm\"uller curves, we also recover results of Mukamel and extend results of Torres-Teigel--Zachhuber establishing that the genus of these Teichm\"uller curves in $\mathcal{H}(2)$ and the Prym loci of $\mathcal{H}(4)$ and $\mathcal{H}(6)$ also go to infinity. The proofs rely on counts of integral points on algebraic hypersurfaces using methods of Bombieri--Pila and Browning--Gorodnik, on counts of orbifold points on Teichm\"uller curves, and on counts of pseudo-Anosov diffeomorphisms with bounded dilatation.