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<p>We introduce the notion of <i>Banach metrics</i> on finitely generated infinite groups. This extends the notion of a Cayley graph (as a metric space). Our motivation comes from trying to detect the existence of virtual homomorphisms into <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>. We show that detection of such homomorphisms through metric functional boundaries of Cayley graphs isn’t always possible. However, we prove that it is always possible to do so through a metric functional boundary of some Banach metric on the group.</p>

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Detecting virtual homomorphisms via Banach metrics

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月19日

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  • Liran Ron-George,
  • Ariel Yadin

摘要

We introduce the notion of Banach metrics on finitely generated infinite groups. This extends the notion of a Cayley graph (as a metric space). Our motivation comes from trying to detect the existence of virtual homomorphisms into \(\mathbb {Z}\) Z . We show that detection of such homomorphisms through metric functional boundaries of Cayley graphs isn’t always possible. However, we prove that it is always possible to do so through a metric functional boundary of some Banach metric on the group.