Relative cohomological dimension of a relatively hyperbolic pair

Abstract We show that the relative cohomological dimension $$\textrm{cd}_R(G,H)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>cd</mml:mtext> <mml:mi>R</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$F_{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> . As a corollary of our methods, we compute $$\textrm{cd}_{\mathbb {Z}}(G,H)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>cd</mml:mtext> <mml:mi>Z</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> in several cases.