Related papers: Graphs of groups and the Atiyah conjecture for one…

This paper has been withdrawn, as it is superseded by arXiv:0806.2122 (Bloch-Kato exponential maps for local fields with imperfect residue fields), which is a more recent version of the same paper.

In 2008, the author proposed a version of duality theory for (not necessarily, Abelian) complex Lie groups, based on the idea of using the Arens-Michael envelope of topological algebra and having an advantage over existing theories in that…

In this note we refine examples by Aka from arithmetic to S-arithmetic groups to show that the vanishing of the $i$-th $\ell^2$-Betti number is not a profinite invariant for all $i \ge 2$.

The paper was removed.

We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete…

In 2019, P. Higgins formulated [1] a question about bipartite graphs (see Conjecture 1 below); this question arises in the study of regular finite semigroups. F. V. Petrov formulated [2] another combinatorial conjecture (Conjecture 3);…

We give a survey on L^2-invariants such as L^2-Betti numbers and L^2-torsion taking an algebraic point of view. We discuss their basic definitions, properties and applications to problems arising in topology, geometry, group theory and…

In 2001 Sir M. F. Atiyah formulated a conjecture C1 and later with P. Sutcliffe two stronger conjectures C2 and C3. These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for any…

We provide some corrections and clarifications to the paper [Gr3] of the title. In particular, we clarify the "left/right" conventions on complex reflection groups and their braid groups. Most importantly, we fill in a gap related to the…

Atiyah proved that the moment map image of the closure of an orbit of a complex torus action is convex. Brion generalized this result to actions of a complex reductive group. We extend their results to actions of a maximal solvable…

I withdraw my paper from arXiv because there is a technical error in the proof of Theorem 1.1. And because of this error, all the results in the paper are untrue. I am very sorry for this.

We unify the known basic theories on $L^2$-Betti numbers and costs in the framework of probability measure preserving discrete groupoids.

Provides a counterexample to a long standing conjecture of A. Adem regarding the behaviour of the integral cohomology of a p-group.

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups…

This paper grew out of an attempt to find a suitable finite sheeted covering of an aspherical 3-manifold so that the cover either has infinite or trivial first homology group. With this motivation we define a new class of groups. These…

The paper is being withdrawn since the results are incorporated in paper arxiv.org/abs/math.AG/0306195.

This paper has been withdrawn by the author due to essential mistakes in some previous versions.

Wolfang L\"uck asked if twisted $L^2$-Betti numbers of a group are equal to the usual $L^2$-Betti numbers rescaled by the dimension of the twisting representation. We confirm this for sofic groups.

This paper has been withdrawn by the author, since it is now a part of "Classical aspects of lightlike dimensional reduction" Class.Quant.Grav.23:7085-7110, 2006, gr-qc/0610011

The paper has been withdrawn.