Trace polynomials for closed curves on the enlarged modular torus: Monotonicity, positivity, and log-concavity

The modular torus is a once-cusped hyperbolic torus with the maximal order of symmetry. The traces of the simple closed geodesics on the modular torus give the geometric version of the classical Markoff numbers. We study trace polynomials for the closed curves on the enlarged modular torus, with the single variable the enlargement factor. We obtain partial ordering of the polynomials for certain simple closed curves. We obtain positivity of the polynomials for all closed curves. We have observed log-concavity for these polynomials and confirmed log-concavity of the polynomials for the simple closed curves. We also mention the problem to enlarge an arbitrary complete one-holed hyperbolic torus. This is joint work with Xiangfei Li.