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<p>To each finitely generated group <i>G</i>, we associate a quasi-isometric invariant called the <i>Dehn spectrum</i> of <i>G</i>. If <i>G</i> is finitely presented, our invariant is closely related to the Dehn function of <i>G</i>, yet provides more information by encoding the isoperimetric behaviour of <i>G</i> at various scales. The main goal of this paper is to initiate the study of the Dehn spectrum of finitely generated (but not necessarily finitely presented) groups. In particular, we compute the Dehn spectrum of small cancellation groups, certain wreath products, and free Burnside groups of sufficiently large odd exponent. We also address some natural questions on the structure of the poset of Dehn spectra. As an application, we show that there exist <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2^{\aleph _0}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <msub> <mi>ℵ</mi> <mn>0</mn> </msub> </msup> </math></EquationSource> </InlineEquation> pairwise non-quasi-isometric finitely generated groups of finite exponent.</p>

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Isoperimetric inequalities in finitely generated groups

  • 期刊论文

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  • 发表日期：

    2026年3月16日

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  • Denis Osin,
  • Ekaterina Rybak

摘要

To each finitely generated group G, we associate a quasi-isometric invariant called the Dehn spectrum of G. If G is finitely presented, our invariant is closely related to the Dehn function of G, yet provides more information by encoding the isoperimetric behaviour of G at various scales. The main goal of this paper is to initiate the study of the Dehn spectrum of finitely generated (but not necessarily finitely presented) groups. In particular, we compute the Dehn spectrum of small cancellation groups, certain wreath products, and free Burnside groups of sufficiently large odd exponent. We also address some natural questions on the structure of the poset of Dehn spectra. As an application, we show that there exist \(2^{\aleph _0}\) 2 ℵ 0 pairwise non-quasi-isometric finitely generated groups of finite exponent.