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This work is motivated by two central questions in the birational geometry ofmoduli spaces of curves -- Fulton's conjecture and the effective cone of $\barM_g$. We study the algebro-geometric aspect of Teichmuller curvesparameterizing square-tiled surfaces with two applications:  (a) there exist infinitely many rigid curves on the moduli space ofhyperelliptic curves, they span the same extremal ray of the cone of movingcurves and their union is Zariski dense, hence they yield infinitely many rigidcurves with the same properties on the moduli space of stable n-pointedrational curves for even n;  (b) the limit of slopes of Teichmuller curves and the sum of Lyapunovexponents of the Hodge bundle determine each other, by which we can have abetter understanding for the cone of effective divisors on the moduli space ofcurves.

Square-tiled surfaces and rigid curves on moduli spaces