Thin buildings

Abstract

Let X be a building of uniform thickness q C 1. L 2 -Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L 2 -cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q . The weighted cohomology makes sense for all real positive values of q , and is computed for small q . If the Davis complex of the Coxeter group is a manifold, a version of Poincaré duality allows to deduce that the L 2 -cohomology of a building with large thickness is concentrated in the top dimension.

WeightedL2–cohomology of Coxeter groups

“…The result of [27] states that, for q 2 R, the L 2 q -homology of † vanishes except in dimension 0. (NB To calculate homology, L 2 q H .…”

“…This number is called the i -th L 2 -Betti number and denoted b i .X I G/. It is proved in [27] (under slightly stronger hypotheses), as well as in Theorem 13.8 of Section 13, that…”

“…We shall use the notation † cc to denote † equipped with this cell structure, where the subscript cc stands for "Coxeter cell." (In [27] this cell structure is denoted † d , where the subscript d stood for "dual cell.") The poset of cells of † cc is W S .…”

“…This extends in a natural way to a measure, also denoted by q , on U .i/ (where U .i/ denotes the entire set of i -cells in U ). As in [27], define the q-weighted L 2 -(co)chains on U (in dimension i ) to be the Hilbert space…”

“…I be a function to some index set I so that i .s/ D i .s 0 / whenever s and s 0 are conjugate. Given an I -tuple q D .q i / i2I of positive real numbers, the second author [27] defined certain "weighted L 2 -cohomology spaces, " here denoted L 2 q H i . †/.…”