Weighted L^2 cohomology of Coxeter groups
by
Michael Davis
Ohio State University
Coauthors: Jan Dymara, Tadeusz Januszkiewicz, Boris Okun

The reduced L²-cohomology of of a CW complex lies somewhere between its ordinary cohomology and its cohomology with compact support. One of the most interesting interpretations of the previous sentence involves the theory of ``weighted L<sup>2</sup>-cohomology'' of the complex X associated to a Coxeter system (W, S). The weight function is of the form q<sup>d</sup>, where d is the combinatorial distance to a base point and q is a positive real number. The corresponding weighted L<sup>2</sup>-cohomology spaces are modules over the ``Hecke - von Neumann algebra'' associated to W with parameter q. Consequently, theses spaces have a von Neumann dimension; the resulting real numbers are called the L²_q- Betti numbers of X. Dymara proved that the L²_q- Betti numbers of X are equal to the L²-Betti numbers of any building of type (W, S) and thickness q+1 (with respect to the von Neumann algebra of a chamber transitive automorphism group G.) As q goes from 0 to infinity, these weighted L²-cohomology spaces interpolate between the ordinary cohomology of X and its cohomology with compact support. There is a precise formulation of this involving the radius of convergence of the growth series of W.

Paper reference: arXiv:math.GT/0402377

Date received: January 5, 2004