The conjugator length function of a finitely generated group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius n in the Cayley graph of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. We prove that polynomials of arbitrary degree arise as conjugator length functions of finitely presented groups. To establish this, we analyse the geometry of conjugation in the discrete model filiform groups <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma _d = \mathbb {Z}^d\rtimes _\phi \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mi>d</mi> </msub> <mo>=</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> <msub> <mo>⋊</mo> <mi>ϕ</mi> </msub> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> is the automorphism of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {Z}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> that fixes the last element of a basis <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$a_1,\dots ,a_d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>d</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and sends <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$a_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$a_ia_{i+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$i<d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo><</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation>. The conjugator length function of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Gamma _d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Γ</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> is polynomial of degree d.

The lengths of conjugators in the model filiform groups

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2026年3月5日

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M. R. Bridson,
T. R. Riley

摘要

The conjugator length function of a finitely generated group $\Gamma $ $Γ$ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius n in the Cayley graph of $\Gamma $ $Γ$ . We prove that polynomials of arbitrary degree arise as conjugator length functions of finitely presented groups. To establish this, we analyse the geometry of conjugation in the discrete model filiform groups $\Gamma _d = \mathbb {Z}^d\rtimes _\phi \mathbb {Z}$ $Γ_{d} = Z^{d} ⋊_{ϕ} Z$ where $\phi $ $ϕ$ is the automorphism of $\mathbb {Z}^d$ $Z^{d}$ that fixes the last element of a basis $a_1,\dots ,a_d$ $a_{1}, \dots, a_{d}$ and sends $a_i$ $a_{i}$ to $a_ia_{i+1}$ $a_{i} a_{i + 1}$ for $i<d$ $i < d$ . The conjugator length function of $\Gamma _d$ $Γ_{d}$ is polynomial of degree d.