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<p>We show that the relative cohomological dimension <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{cd}_R(G,H)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>cd</mtext> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a relatively hyperbolic pair (<i>G</i>,&#xa0;<i>H</i>) is always finite when <i>G</i> does not contain <i>R</i>-torsion. We also show that this dimension is preserved under quasi-isometries, provided that <i>G</i> is torsion-free and the peripheral subgroup <i>H</i> is unconstricted and of type <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(F_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>. As a corollary of our methods, we compute <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textrm{cd}_{\mathbb {Z}}(G,H)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>cd</mtext> <mi mathvariant="double-struck">Z</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in several cases.</p>

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Relative cohomological dimension of a relatively hyperbolic pair

  • 期刊论文

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  • 发表日期：

    2026年2月26日

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  • Harsh Patil

摘要

We show that the relative cohomological dimension \(\textrm{cd}_R(G,H)\) cd R ( G , H ) of a relatively hyperbolic pair (G, H) is always finite when G does not contain R-torsion. We also show that this dimension is preserved under quasi-isometries, provided that G is torsion-free and the peripheral subgroup H is unconstricted and of type \(F_{\infty }\) F ∞ . As a corollary of our methods, we compute \(\textrm{cd}_{\mathbb {Z}}(G,H)\) cd Z ( G , H ) in several cases.