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Curvature criterion for vanishing of group cohomology

Zohar Grinbaum-Reizis, Izhar Oppenheim

Abstract

We introduce a new geometric criterion for vanishing of cohomology for BN-pair groups. In particular, this new criterion yields a sharp vanishing of cohomology result for all BN-pair groups acting on non-thin affine building.

Curvature criterion for vanishing of group cohomology

Abstract

We introduce a new geometric criterion for vanishing of cohomology for BN-pair groups. In particular, this new criterion yields a sharp vanishing of cohomology result for all BN-pair groups acting on non-thin affine building.

Paper Structure

This paper contains 5 sections, 21 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Decomposition Theorem in Hilbert spaces
  3. Vanishing of Cohomology for groups acting on simplicial complexes
  4. Vanishing of cohomology for groups acting on buildings
  5. Equivariant embedding interpretation of our criterion

Key Result

Theorem \oldthetheorem

Let $G$ be a BN-pair group acting on a building $X$ such that $X$ is $n$-dimensional with $n \geq 2$ and all the $1$-dimensional links of $X$ are finite. Denote $C$ to be the cosine matrix of the Coxeter system associated with the Coxeter group that arises from the BN-pair of $G$ and $\widetilde{\mu

Theorems & Definitions (53)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Definition \oldthetheorem: The cosine matrix of $X$
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem: Angle between subspaces
  • ...and 43 more