Given a graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> and a number n, the associated <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n^{th}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>n</mi> <mrow> <mi mathvariant="italic">th</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> graph braid group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$B_n(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the fundamental group of the unordered configuration space of n points on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. Świątkowski showed that for a given <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> and n large enough, there is a free abelian subgroup of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$B_n(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of rank equal to the cohomological dimension of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$B_n(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this note, we observe that at the cost of possibly adding additional points, we can find a subgroup of the same cohomological dimension which is a direct product of non-abelian free groups, and give some applications. In particular, we show that the topological complexity of every graph braid group stabilizes for large enough n.

Products of free groups inside graph braid groups

期刊论文
发表日期：

2025年10月16日

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Kasia Jankiewicz,
Kevin Schreve

摘要

Given a graph $\Gamma $ $Γ$ and a number n, the associated $n^{th}$ $n^{th}$ graph braid group $B_n(\Gamma )$ $B_{n} (Γ)$ is the fundamental group of the unordered configuration space of n points on $\Gamma $ $Γ$ . Świątkowski showed that for a given $\Gamma $ $Γ$ and n large enough, there is a free abelian subgroup of $B_n(\Gamma )$ $B_{n} (Γ)$ of rank equal to the cohomological dimension of $B_n(\Gamma )$ $B_{n} (Γ)$ . In this note, we observe that at the cost of possibly adding additional points, we can find a subgroup of the same cohomological dimension which is a direct product of non-abelian free groups, and give some applications. In particular, we show that the topological complexity of every graph braid group stabilizes for large enough n.