McMullen's expansion conjecture for arithmetic Veech groups
For a translation surface in the stratum , the associated Veech group is arithmetic when its orbit gives rise to the graphs under consideration. McMullen's expansion conjecture. The family of graphs associated to arithmetic Veech groups in is an expander family. Equivalently, the spectral gaps of the arithmetic Veech groups possess a uniform lower bound. This conjecture connects effective density near Teichmüller curves with uniform spectral expansion; the source does not state whether it has been resolved.
References
Primary source
Siyuan Tang, “Effective density of surfaces near Teichmüller curves”, arXiv:2411.03560 (2025).
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