Cohomology of Coxeter groups with group ring coefficients: II

Abstract

For any Coxeter group W , we define a filtration of H * (W; ZW) by W -submodules and then compute the associated graded terms. More generally, if U is a CW complex on which W acts as a reflection group we compute the associated graded terms for H * (U) and, in the case where the action is proper and cocompact, for H * c (U).

Compactly supported cohomology of buildings

“…As we have said, special cases of the Decomposition Theorem were proved in [10]. The method of [10] was simply to find a basis for A adapted to its decomposition into the O A T .…”

“…The method of [10] was simply to find a basis for A adapted to its decomposition into the O A T . In the general case, finding an explicit description of such a basis is problematic.…”

“…The corresponding cohomology groups were studied in [10], [7]. In particular, when X D K, these cohomology groups are isomorphic to H .W I ZW /.…”

“…In the case whereˆD W , this was done in [7], [10], [13]. For a general (thick) building, in the case where .W; S/ is right-angled, it was done in [10], Theorem 6.6. It was claimed in full generality in [13]; however, there is a mistake in the proof (see [14]).…”

“…Let O A T be a complementary summand. As in [10], [11], [16] the main computation is a consequence of the following Decomposition Theorem (proved as Theorem 7.1 in Section 7).…”