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) SL ( 2 , Z ) -orbits of primitive origamis in the stratum H(2) 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) H ( 2 ) is generalised and reused in Lanneau and Nguyen’s classification of the orbits of Prym eigenforms in H(4) H ( 4 ) and H(6) 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(N2/3log N) O ( N 2 / 3 log N ) , where N is the size of the orbit graph.