Tadeusz Januszkiewicz

Fundamental groups of blow-ups

Abstract Many examples of nonpositively curved closed manifolds arise as real blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W and if the blow-up locus is W -invariant, then the resulting manifold will admit a cell decomposition whose maximal cells are all combinatorially isomorphic to a given convex polytope P . In other words, M admits a tiling with tile P . The universal covers of such examples yield tilings of R n whose symmetry groups are generated by involutions but are not, in general, reflection groups. We begin a study of these “mock reflection groups”, and develop a theory of tilings that includes the examples coming from blow-ups and that generalizes the corresponding theory of reflection tilings. We apply our general theory to classify the examples coming from blow-ups in the case where the tile P is either the permutohedron or the associahedron.

Weighted

Given a Coxeter system .W;S/ and a positive real multiparameter q, we study the “weighted L 2 ‐cohomology groups,” of a certain simplicial complex † associated to .W;S/. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to .W;S/ and the multiparameter q. They have a “von Neumann dimension” with respect to the associated “Hecke‐von Neumann algebra” Nq. The dimension of the i‐th cohomology group is denoted b i .†/. It is a nonnegative real number which varies continuously with q. When q is integral, the b i .†/ are the usual L 2 ‐Betti numbers of buildings of type .W;S/ and thickness q. For a certain range of q, we calculate these cohomology groups as modules over Nq and obtain explicit formulas for the b i .†/. The range of q for which our calculations are valid depends on the region of convergence of the growth series of W . Within this range, we also prove a Decomposition Theorem for Nq, analogous to a theorem of LSolomon on the decomposition of the group algebra of a finite Coxeter group.

L^2

We dene graph products of families of pairs of groups and study the question when two such graph products are commensurable. As an application we prove linearity of certain graph products. AMS Classication 20F65; 57M07

–cohomology of Coxeter groups

We compute the` 2 -Betti numbers of the complement of a finite collection of affine hyperplanes in C n . At most one of the` 2 -Betti numbers is nonzero.

Commensurability of graph products

In this note we give the quasi-isometry classification for a class of right angled Artin groups. In particular, we obtain the first such classification for a class of Artin groups with dimension larger than 2; our families exist in every dimension.

The ` 2 -cohomology of hyperplane complements

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).

Quasi-isometric classification of some high dimensional right-angled Artin groups

We construct finitely generated groups with strong fixed point properties. Let Xac be the class of Hausdorff spaces of finite covering dimension which are mod–p acyclic for at least one prime p. We produce the first examples of infinite finitely generated groups Q with the property that for any action of Q on any X ?Xac, there is a global fixed point. Moreover, Q may be chosen to be simple and to have Kazhdan’s property (T). We construct a finitely presented infinite group P that admits no nontrivial action on any manifold in Xac. In building Q, we exhibit new families of hyperbolic groups: for each n ? 1 and each prime p, we construct a nonelementary hyperbolic group Gn,p which has a generating set of size n + 2, any proper subset of which generates a finite p–group.

Cohomology of Coxeter groups with group ring coefficients: II

We compute the compactly supported cohomology of the standard realization of any locally finite building. MathematicsSubjectClassification(2010). Primary 20F65; Secondary 20E42, 20F55, 20J06, 57M07.

Infinite groups with fixed point properties

We construct examples of 4-dimensional manifolds M supporting a locally CAT(0)metric, whose universal covers Q M satisfy Hruska’s isolated flats condition, and contain 2-dimensional flats F with the property that @1F S S ,! S S @1 Q M are nontrivial knots. As a consequence, we obtain that the group 1.M/ cannot be isomorphic to the fundamental group of any compact Riemannian manifold of nonpositive sectional curvature. In particular, if K is any compact locally CAT(0)-manifold, then M K is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.

Compactly supported cohomology of buildings

First published in Transactions of the American Mathematical Society in volume 362 and issue 1, published by the American Mathematical Society.