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In 2018, Kielak gave a virtual fibering criterion for RFRS groups. In this paper, we present a simpler proof of this., Comment: 17 pages

In this paper we define the coarse (co)homology of the complement of a subspace in a metric space, generalizing the coarse (co)homology of Roe. We give a model space which encodes coarse geometric structure of the complement. We also introduce a new approach to coarse Poincar\'e duality spaces. We prove a version of coarse Alexander duality for these spaces and give a homological criterion for a space to be a coarse PD($n$) space. Our approach is inspired by the work of Kapovich and Kleiner, but is somewhat different, and we believe, simpler.

We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and -- for almost all primes $p$ -- $\mathbb{F}_p$-homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic $7$-manifolds that do not virtually fiber over a circle out of graph products of large finite groups., Comment: Fixed a gap in Section 10, added Theorem 5.4, numerous small fixes and changes

Suppose a residually finite group $G$ acts cocompactly on a contractible complex with strict fundamental domain $Q$, where the stabilizers are either trivial or have normal $\mathbb{Z}$-subgroups. Let $\partial Q$ be the subcomplex of $Q$ with nontrivial stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of $G$. We show that independent of the chain, the normalized torsion limits to the torsion of $\partial Q$, shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual $L^2$-torsion of $G$ in terms of the torsion of stabilizers and topology of $\partial Q$. In particular, we get complete answers for right-angled Artin groups, which shows they satisfy a torsion analogue of the L\"uck approximation theorem., Comment: Minor updates and fixes

We compute the mod $p$ homology growth of residual sequences of finite index normal subgroups of right-angled Artin groups. We find examples where this differs from the rational homology growth, which implies the homology of subgroups in the sequence has lots of torsion. More precisely, the homology torsion grows exponentially in the index of the subgroup. For odd primes $p$, we construct closed locally CAT(0) manifolds with nonzero mod $p$ homology growth outside the middle dimension. These examples show that Singer's conjecture on rational homology growth and L\"uck's conjecture on torsion homology growth are incompatible with each other, so at least one of them must be wrong., Comment: Added Corollary 10, minor changes

Let $X$ be a CAT(0) cubical complex. The growth series of $X$ at $x$ is $G_{x}(t)=\sum_{y \in Vert(X)} t^{d(x,y)}$, where $d(x,y)$ denotes $\ell_{1}$-distance between $x$ and $y$. If $X$ is cocompact, then $G_{x}$ is a rational function of $t$. In the case when $X$ is the Davis complex of a right-angled Coxeter group it is a well-known that $G_{x}(t)=1/f_{L}(-t/(1+t))$, where $f_{L}$ denotes the $f$-polynomial of the link $L$ of a vertex of $X$. We obtain a similar formula for general cocompact $X$. We also obtain a simple relation between the growth series of individual orbits and the $f$-polynomials of various links. In particular, we get a simple proof of reciprocity of these series ($G_{x}(t)=\pm G_{x}(t^{-1})$) for an Eulerian manifold $X$., Comment: 8 pages

The action dimension of a discrete group $\Gamma$ is the smallest dimension of a contractible manifold which admits a proper action of $\Gamma$. Associated to any flag complex $L$ there is a right-angled Artin group, $A_L$. We compute the action dimension of $A_L$ for many $L$. Our calculations come close to confirming the conjecture that if an $\ell^2$-Betti number of $A_L$ in degree $l$ is nonzero, then the action dimension of $A_L$ is $\ge 2l$., Comment: 16 pages

We study group actions on manifolds that admit hierarchies, which generalizes the idea of Haken n-manifolds introduced by Foozwell and Rubinstein. We show that these manifolds satisfy the Singer conjecture in dimensions $n \le 4$. Our main application is to Coxeter groups whose Davis complexes are manifolds; we show that the natural action of these groups on the Davis complex has a hierarchy. Our second result is that the Singer conjecture is equivalent to the cocompact action dimension conjecture, which is a statement about all groups, not just fundamental groups of closed aspherical manifolds., Comment: 11 pages

We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W, replacing the group von Neumann algebra NW with the Hecke-von Neumann algebra N_qW., Comment: 25 pages

We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups., Comment: Minor changes

We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of the hyperplane arrangement., Comment: Second version corrects some misprints in the definition of U_G

We compute: * the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), Bestvina-Brady groups, and graph products of groups, * the L^2-Betti numbers of Bestvina-Brady groups and of graph products of groups, * the weighted L^2-Betti numbers of graph products of Coxeter groups. In the case of arbitrary graph products there is an additional proviso: either all factors are infinite or all are finite.(However, for graph products of Coxeter groups this proviso is unnecessary.), Comment: 56 pages

We compute the compactly supported cohomology of the standard realization of any locally finite building.

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)., Comment: This is the version published by Algebraic & Geometric Topology on 15 September 2006

Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a contractible cubical complex Sigma_L (the Davis complex) on which W_L acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a generalized homology sphere, then Sigma_L is a contractible homology manifold. We prove a generalized version of the Singer Conjecture (on the vanishing of the reduced weighted L^2_q-cohomology above the middle dimension) for the right-angled Coxeter groups based on barycentric subdivisions in even dimensions. We also prove this conjecture for the groups based on the barycentric subdivision of the boundary complex of a simplex., Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper28.abs.html

We construct series of examples of exotic smooth structures on compact locally symmetric spaces of noncompact type. In particular, we obtain higher rank examples, which do not support Riemannian metric of nonpositive curvature. The examples are obtained by taking the connected sum with an exotic sphere. To detect the change of the smooth structure we use a tangential map from the locally symmetric space its dual compact type twin., Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-18.abs.html

We construct a tangential map from a locally symmetric space of noncompact type to its dual compact type twin. By comparing the induced map in cohomology to a map defined by Matsushima, we conclude that in the equal rank case the map has a nonzero degree., Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-35.abs.html

Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a cubical complex \Sigma_L on which W_L acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of \Sigma_L is L and (2) \Sigma_L is contractible. It follows that if L is a triangulation of S^{n-1}, then \Sigma_L is a contractible n-manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced L^2-homology except in the middle dimension) in the case of \Sigma_L where L is a triangulation of S^{n-1}. The program succeeds when n < 5. This implies the Charney-Davis Conjecture on flag triangulations of S^3. It also implies the following special case of the Hopf-Chern Conjecture: every closed 4-manifold with a nonpositively curved, piecewise Euclidean, cubical structure has nonnegative Euler characteristic. Our methods suggest the following generalization of the Singer Conjecture. Conjecture: If a discrete group G acts properly on a contractible n-manifold, then its L^2-Betti numbers b_i^{(2)} (G)$ vanish for i>n/2., Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper2.abs.html

In this article, we generalize Eberlein's Rigidity Theorem to the singular case, namely, one of the spaces is only assumed to be a CAT(0) topological manifold. As a corollary, we get that any compact irreducible but locally reducible locally symmetric space of noncompact type does not admit a nonpositively curved (in the Aleksandrov sense) piecewise Euclidean structure. Any hyperbolic manifold, on the other hand, does admit such a structure., Comment: 28 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper13.abs.html