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<p>In (Cuntz and Kühne 2025), Cuntz and Kühne introduced a particular class of hyperplane arrangements stemming from a given graph, so called <i>connected subgraph arrangements</i>. In this note we strengthen some of the result from (Cuntz and Kühne 2025) and prove new ones for members of this class. For instance, we show that aspherical members within this class stem from a rather restricted set of graphs. Specifically, if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathscr {A}}_G\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> is an aspherical connected subgraph arrangement, then <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathscr {A}}_G\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> is free with the unique possible exception when the underlying graph <i>G</i> is the complete graph on 4 nodes.</p>

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On connected subgraph arrangements

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年6月23日

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  • Lorenzo Giordani,
  • Tilman Möller,
  • Paul Mücksch,
  • Gerhard Röhrle

摘要

In (Cuntz and Kühne 2025), Cuntz and Kühne introduced a particular class of hyperplane arrangements stemming from a given graph, so called connected subgraph arrangements. In this note we strengthen some of the result from (Cuntz and Kühne 2025) and prove new ones for members of this class. For instance, we show that aspherical members within this class stem from a rather restricted set of graphs. Specifically, if \({\mathscr {A}}_G\) A G is an aspherical connected subgraph arrangement, then \({\mathscr {A}}_G\) A G is free with the unique possible exception when the underlying graph G is the complete graph on 4 nodes.