Related papers: Free Lamplighter Groups and a Question of Atiyah

We prove that there are examples of finitely generated groups G together with group ring elements Q \in \bbQ G for which the von Neumann dimension \dim_{LG}\ker Q is irrational, so (in conjunction with other known results) answering a…

Let $\Gamma$ be a discrete group. Following Linnell and Schick one can define a continuous ring $c(\Gamma)$ associated with $\Gamma$. They proved that if the Atiyah Conjecture holds for a torsion-free group $\Gamma$, then $c(\Gamma)$ is a…

The so-called Atiyah conjecture states that the von Neumann dimensions of the L2-homology modules of free G-CW-complexes belong to a certain set of rational numbers, depending on the finite subgroups of G. In this article we extend this…

We consider noncommutative rational functions as well as matrices in polynomials in noncommuting variables in two settings: in an algebraic context the variables are formal variables, and their rational functions generate the "free field";…

In this paper, we study a family of finitely generated elementary amenable iet-groups. These groups are generated by finitely many rationals iets and rotations. For them, we state criteria for not virtual nilpotency or solvability, and we…

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely…

We use elementary methods to compute the L2-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the…

Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of C that is closed under complex conjugation. Let U(G) denote…

A group G has homological dimension less or equal to 1 if it is locally free. We prove the converse provided that G satisfies the Atiyah Conjecture about L^2-Betti numbers. We also show that a finitely generated elementary amenable group G…

Let G be a group and let K be a field of characteristic zero. We shall prove that KG can be embedded into a von Neumann unit-regular ring. In the course of the proof, we shall obtain a result relevant to the Atiyah conjecture.

Given a finite group $H$ and a free group $F_n$, we prove that the wreath product $H\wr F_n$ admits a metrically proper, isometric action on a Hilbert space.

In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group $F$, the lamplighter…

We study line patterns in a free group by considering the topology of the decomposition space, a quotient of the boundary at infinity of the free group related to the line pattern. We show that the group of quasi-isometries preserving a…

We describe a coordinate-free perspective on conformal nets, as functors from intervals to von Neumann algebras. We discuss an operation of fusion of intervals and observe that a conformal net takes a fused interval to the fiber product of…

We define homological area-radius pairs with surface diagrams. Using these, we adapt a proof of Gersten and Short \cite{gersten2002} to obtain a homological isoperimetric inequality for subgroups of type $FP_2$ which appear as kernels of…

We establish several properties of the free Stein dimension, an invariant for finitely generated unital tracial $*$-algebras. We give formulas for its behaviour under direct sums and tensor products with finite dimensional algebras. Among a…

Let $X_1,\dots,X_n$ be operators in a finite von Neumann algebra and consider their division closure in the affiliated unbounded operators. We address the question when this division closure is a skew field (aka division ring) and when it…

We show that the reduced von Neumann algebras of the free orthogonal and free unitary quantum groups have the Haagerup approximation property. Using this result and a Haagerup-type inequality for these quantum groups due to Vergnioux (J.…

For a self-symmetric tracial von Neumann algebra $A$, we study rescalings of $A^{*n} * L\mathbb{F}_r$ for $n \in \mathbb{N}$ and $r \in (1, \infty]$ and use them to obtain an interpolation $\mathcal{F}_{s,r}(A)$ for all real numbers $s>0$…

We prove that, if a discrete group $G$ is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of $G$ is non-amenable with respect to the topology generated by its rank metric. This…