Diameter bounds for $$\varvec{\text {SL}(2,\mathbb {Z})}$$-orbits of origamis in $$\varvec{\mathcal {H}(2)}$$ and the Prym loci in $$\varvec{\mathcal {H}(4)}$$ and $$\varvec{\mathcal {H}(6)}$$

Abstract Using algorithms implicit in the classification of $$\text {SL}(2,\mathbb {Z})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SL</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -orbits of primitive origamis in the stratum $$\mathcal {H}(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> due to Hubert–Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from $$\mathcal {H}(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is generalised and reused in Lanneau and Nguyen’s classification of the orbits of Prym eigenforms in $$\mathcal {H}(4)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$\mathcal {H}(6)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mn>6</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , 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 $$O(N^{2/3}\log N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>log</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where N is the size of the orbit graph.