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February 28, 2026Journal of Homotopy and Related Structures0 citationsOpen Access

Relative cohomological dimension of a relatively hyperbolic pair

HPHarsh PatilUniversity of Bristol

Key Points

  • The aim is to explore the relative cohomological dimension of hyperbolic pairs and its preservation properties.
  • Analyzed the relative cohomological dimension of pairs (G, H)
  • Examined conditions under which this dimension is finite
  • Investigated the effects of quasi-isometries on dimension preservation
  • Computed specific cases for cd_{Z}(G,H) using derived methods
  • Confirmed that cd_{R}(G,H) is finite when G is torsion-free
  • Showed that relative cohomological dimension is preserved under quasi-isometries
  • Calculated cd_{Z}(G,H) for multiple cases, demonstrating consistency

Abstract

Abstract We show that the relative cohomological dimension cdR (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 F ∞. As a corollary of our methods, we compute cd ₙ (G, H) cd Z (G, H) in several cases.

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Cite This Study

Harsh Patil (2026) studied this question.

synapsesocial.com/papers/69a2877b0a974eb0d3c03402https://doi.org/10.1007/s40062-026-00403-1
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