<p>Using algorithms implicit in the classification of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\text {SL}(2,\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SL</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-orbits of primitive origamis in the stratum <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {H}(2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> due to Hubert–Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {H}(2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is generalised and reused in Lanneau and Nguyen’s classification of the orbits of Prym eigenforms in <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {H}(4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathcal {H}(6)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$O(N^{2/3}\log N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow> <mn>2</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> <mo>log</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <i>N</i> is the size of the orbit graph.</p>

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 $\varvec{\mathcal {H}(6)}$

期刊论文
出版模式：

开放获取
发表日期：

2026年3月26日

查看全文

Luke Jeffreys,
Carlos Matheus

摘要

Using algorithms implicit in the classification of $\text {SL}(2,\mathbb {Z})$ $SL (2, Z)$ -orbits of primitive origamis in the stratum $\mathcal {H}(2)$ $H (2)$ due to Hubert–Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from $\mathcal {H}(2)$ $H (2)$ is generalised and reused in Lanneau and Nguyen’s classification of the orbits of Prym eigenforms in $\mathcal {H}(4)$ $H (4)$ and $\mathcal {H}(6)$ $H (6)$ , 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 $O(N^{2/3}\log N)$ $O (N^{2 / 3} log N)$ , where N is the size of the orbit graph.

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 \(\varvec{\mathcal {H}(6)}\)

摘要