Related papers: C*-superrigidity of 2-step nilpotent groups

We give a uniform construction that, on input of a recursive presentation $P$ of a group, outputs a recursive presentation of a torsion-free group, isomorphic to $P$ whenever $P$ is itself torsion-free. We use this to re-obtain a known…

We construct discrete groups $G$ with infinite center that are nevertheless W*-superrigid, meaning that the group von Neumann algebra $L(G)$ fully remembers the group $G$. We obtain these rigidity results both up to isomorphisms and up to…

In the theory of C*-algebras, interesting noncommutative structures arise as deformations of the tensor product. For instance, the rotation algebra may be seen as a scalar twist deformation of the tensor product of the functions on the…

Plante-Thurston proved that every nilpotent subgroup of $\Diff^2(S^1)$ is abelian. One of our main results is a sharp converse: $\Diff^1(S^1)$ contains every finitely-generated, torsion-free nilpotent group.

In this paper we study the part of the $K$-theory of the reduced $C^*$-algebra arising from torsion elements of the group, and in particular we study the pairing of $K$-theory with traces and when traces can detect certain $K$-theory…

We classify finite-dimensional Nichols algebras over finite nilpotent groups of odd order in group-theoretical terms. The main step is to show that the conjugacy classes of such finite groups are either abelian or of type C; this property…

Anabelian geometry suggests that, for suitably geometric objects, their \'etale fundamental groups determine the geometric objects up to isomorphism. From a group-theoretic viewpoint, this philosophy requires rigidity properties, which…

We show that if a torsion free nilpotent group $G$ has a balanced presentations and Hirsch length $h(G)>3$ then $\beta_1(G;\mathbb{Q})=2$. There is just one such group which is torsion-free and of Hirsch length $h=4$, and none with $h=5$.…

For any nilpotent Lie group $G$ we provide a description of the image of its $C^*$-algebra through its operator-valued Fourier transform. Specifically, we show that $C^*(G)$ admits a finite composition series such that that the spectra of…

Relying on the techniques and ideas from our recent paper [13], we prove several anti-classification results for various rigidity conditions in countable abelian and nilpotent groups. We prove three main theorems: (1) the rigid abelian…

We develop a theory of type semigroups for arbitrary twisted, not necessarily Hausdorff \'etale groupoids. The type semigroup is a dynamical version of the Cuntz semigroup. We relate it to traces, ideals, pure infiniteness, and stable…

In this paper we will show that finitely generated torsion-free 2-step nilpotent groups of Hirsch length at most 6 do not have the $R_{\infty}$-property, while there are examples of such groups of Hirsch length 7 that do have the…

Using algebraic and topological K-theory together with complex C^*-algebras, we prove that every abelian group may be realized as the centre of a strongly torsion generated group whose integral homology is zero in dimension one and…

Let $\mathcal{T}$ denote the class of finitely generated torsion-free nilpotent groups. For a group $G$ let $F(G)$ be the set of isomorphism classes of finite quotients of $G$. Pickel proved that if $G \in \mathcal{T}$, then the set…

It is proved that that every nilpotent contraction in a quotient C*-algebra can be lifted to a nilpotent contraction. As a consequence we get that the universal C*-algebra generated by a nilpotent contraction is projective. This answers the…

In the paper we study irreducible representations of some nilpotent groups of finite abelian total rank. The main result of the paper states that if a torsion-free minimax group $G$ of nilpotency class 2 admits a faithful irreducible…

We study the class of pseudocompact C*-algebras, which are the logical limits of finite-dimensional C*-algebras. The pseudocompact C*-algebras are unital, stably finite, real rank zero, stable rank one, and tracial. We show that the…

We prove that a subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F.

In this paper, the $C^*$-algebra of the seven-dimensional un-decomposable nilpotent Lie group is characterized explicitly for the first time(see \cite{chin}). Furthermore, the topology of its spectrum is described as a preparation for the…

A residually nilpotent group is \emph{$k$-parafree} if all of its lower central series quotients match those of a free group of rank $k$. Magnus proved that $k$-parafree groups of rank $k$ are themselves free. In this note we mimic this…