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A vanishing theorem for the homology of discrete subgroups of $\mathrm{Sp}(n,1)$ and $\mathrm{F}_4^{-20}$
Geometric Topology · Submitted 11 Jun 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For any discrete, torsion-free subgroup $\Gamma$ of $\mathrm{Sp}(n,1)$ (resp.\ $\mathrm{F}_4^{-20}$) with no parabolic elements, we prove that $H_{4n-1}(\Gamma;V)=0$ (resp.\ $H_i(\Gamma;V)=0$ for $i=13,14,15$) for any $\Gamma$--module $V$.
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For any discrete, torsion-free subgroup $\Gamma$ of $\mathrm{Sp}(n,1)$ (resp.\ $\mathrm{F}_4^{-20}$) with no parabolic elements, we prove that $H_{4n-1}(\Gamma;V)=0$ (resp.\ $H_i(\Gamma;V)=0$ for $i=13,14,15$) for any $\Gamma$--module $V$. The main technical advance is a new bound on the $p$--Jacobian of the barycenter map of Besson--Courtois--Gallot. We also apply this estimate to obtain an inequality between the critical exponent and homological dimension of $\Gamma$, improving on work of M.~Kapovich.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For any discrete, torsion-free subgroup $\Gamma$ of $\mathrm{Sp}(n,1)$ (resp.\ $\mathrm{F}_4^{-20}$) with no parabolic elements, we prove that $H_{4n-1}(\Gamma;V)=0$ (resp.\ $H_i(\Gamma;V)=0$ for $i=13,14,15$) for any $\Gamma$--module $V$. | result 0.99 | 0.96 | 0.94 | asserted 1.00 |
| s1 | The main technical advance is a new bound on the $p$--Jacobian of the barycenter map of Besson--Courtois--Gallot. | contribution 0.75 | 0.92 | 0.04 | asserted 1.00 |
| s2 | We also apply this estimate to obtain an inequality between the critical exponent and homological dimension of $\Gamma$, improving on work of M.~Kapovich. | result 0.60 | 0.94 | 0.05 | asserted 1.00 |
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Original abstract
For any discrete, torsion-free subgroup $\Gamma$ of $\mathrm{Sp}(n,1)$ (resp.\ $\mathrm{F}_4^{-20}$) with no parabolic elements, we prove that $H_{4n-1}(\Gamma;V)=0$ (resp.\ $H_i(\Gamma;V)=0$ for $i=13,14,15$) for any $\Gamma$--module $V$. The main technical advance is a new bound on the $p$--Jacobian of the barycenter map of Besson--Courtois--Gallot. We also apply this estimate to obtain an inequality between the critical exponent and homological dimension of $\Gamma$, improving on work of M.~Kapovich.
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Cites as background 2 papers
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Recently, Kim–Kim [ref] and Lafont–Wang [ref] gave general estimates for $p$–Jacobians in all real ranks.
Simplicial volume, Barycenter method, and Bounded cohomology · Estimating the $p$–Jacobian · 0.86
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Recently, Kim–Kim [ref] and Lafont–Wang [ref] gave general estimates for $p$–Jacobians in all real ranks.
Barycentric straightening and bounded cohomology · Estimating the $p$–Jacobian · 0.86
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