Diameter bounds for SL(2,Z)-orbits of origamis in H(2) and the Prym loci in H(4) and H(6)

Using algorithms implicit in the classification of SL(2,Z)-orbits of primitive origamis in the stratum H(2) due to Hubert–Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from H(2) is generalised and reused in Lanneau and Nguyen’s classification of the orbits of Prym eigenforms in H(4) and H(6), we are also able to obtain diameter bounds for the orbit graphs in this setting as well. In each stratum, we obtain diameter bounds of the form O(N2/3logN), where N is the size of the orbit graph.