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We establish new metric characterizations for the norm (respectively, ultraweak) closure of the convex hull of a bounded set in an arbitrary $C^*$-algebra (respectively, von Neumann algebra), and provide applications of these results to the majorization theory.

A sequence $a_0

We introduce an Ulam-type stability condition for positive definite maps defined on a countable group and prove that this condition characterizes amenability.

We study the distribution of roots of rank 2 in nonpositively curved 2-complexes with Moebius--Kantor links. For every face in such a complex, the parity of the number of roots of rank 2 in a neighbourhood of the face is a well-defined geometric invariant determined by the root distribution. We study the relation between the root distribution and the parity distribution. We prove that there exist parity distributions in flats which are disallowed in Moebius--Kantor complexes. This contrasts with the fact that every root distribution can be realized. We classify the root distributions associated with an even parity distribution (i.e., such that every face is even) on a flat plane. We prove that there exists up to isomorphism a unique even simply connected Moebius--Kantor complex -- namely, the Pauli complex.

We describe conditions that characterize amenability for groups in terms of positive definite functions valued in a von Neumann algebra.

We study a geometric construction of certain finite index subgroups of Aut(F2).

We study a family of tessellations of the Euclidean plane which are obtained by local developments of algebraic equations satisfied by the Pauli matrices.

We study a geometric action on a CAT(0) space of a finite index subgroup of the quotient group of the braid group on 4 strands by its center.

We study the isomorphism types of simply connected complexes of rank 7/4 using a local invariant called the parity. We show that the parity can be computed explicitly in certain constructions arising from surgery.

We construct a model of random groups of rank 7/4, and show that in this model the random group has the exponential mesoscopic rank property.

We study constructions of groups, in particular of groups of intermediate rank, which are accessible to surgery techniques., Comment: updated reference list

This paper studies Aut(F2) and groups closely related to it from a geometric perspective.

We define entropy invariants for abstract algebraic structures using an asymptotic Boltzmann formula.

We consider models of random groups in which the typical group is of intermediate rank (in particular, it is not hyperbolic). These models are parallel to M. Gromov's well-known constructions and include for example a "density model" for groups of intermediate rank. The main novelty is the higher rank nature of the random groups. They are randomization of certain families of lattices in algebraic groups (of rank 2) over local fields.

It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability measure preserving $s$-regular ergodic discrete groupoids amalgamated over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$ satisfies the equality \[ s(G)=\h(G_1^0)s(G_1)+\h(G_2^0)s(G_2)-\h(G_3^0)s(G_3) \] where $\h$ is the normalized Haar measure on $G$.

We show that the group of presentation $< a,b,c,s,t\mid c=ab=ba,\, c^2=sas^{-1}=tbt^{-1}>$ (introduced by D. Wise) has the property of rapid decay.

We prove the Haagerup property for an infinite discrete group constructed using surgery on a Euclidean Tits building of type $\tilde A_2$.

For discrete measured groupoids preserving a probability measure we introduce a notion of sofic dimension that measures the asymptotic growth of the number of sofic approximations on larger and larger finite sets. In the case of groups we give a formula for free products with amalgamation over an amenable subgroup. We also prove a free product formula for measure-preserving actions., Comment: 38 pages, minor changes and corrections

Given an ergodic probability measure preserving dynamical system $\G\acts (X,\mu)$, where $\G$ is a finitely generated countable group, we show that the asymptotic growth of the number of finite models for the dynamics, in the sense of sofic approximations, is an invariant of orbit equivalence. We then prove an additivity formula for free products with amenable (possibly trivial) amalgamation. In particular, we obtain purely combinatorial proofs of several results in orbit equivalence theory.

In this paper, we show how to construct examples of closed manifolds with explicitly computed irrational, even transcendental L2 Betti numbers, defined via the universal covering. We show that every non-negative real number shows up as an L2-Betti number of some covering of a compact manifold, and that many computable real numbers appear as an L2-Betti number of a universal covering of a compact manifold (with a precise meaning of computable given below). In algebraic terms, for many given computable real numbers (in particular for many transcendental numbers) we show how to construct a finitely presented group and an element in the integral group ring such that the L2-dimension of the kernel is the given number. We follow the method pioneered by Austin in "Rational group ring elements with kernels having irrational dimension" arXiv:0909.2360) but refine it to get very explicit calculations which make the above statements possible., Comment: 25 pages; v2: small corrections and improvements of the presentation. Final version to appear in Journal of the London Mathematical Society

We introduce and study a family of countable groups constructed from Euclidean buildings by "removing" suitably chosen subsets of chambers.

For a type II_1 ergodic measured equivalence relation R on a probability space without atom, we prove that h(R)=2C(R)-2, where C(R) is the cost, and h(R) the isoperimetric constant. This follows recent result by Lyons and the authors., Comment: 5 pages

We prove that the braid group $B_4$ on 4 strings, as well as its central quotient $B_4/< z>$, have the property RD of Haagerup-Jolissaint. It follows that the automorphism group $\Aut(F_2)$ of the free group $F_2$ on 2 generators has property RD. We also prove that the braid group $B_4$ is a group of intermediate rank (of dimension 3). Namely, we show that both $B_4$ and its central quotient have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats., Comment: reference added, minor corrections

We establish criteria for turbulence in certain spaces of C*-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X,\mu) and on the hyperfinite II_1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving flows on (X,\mu) is generically turbulent., Comment: 27 pages

It is shown that $2\beta_1(\G)\leq h(\G)$ for any countable group $\G$, where $\beta_1(\G)$ is the first $\ell^2$-Betti number and $h(\G)$ the uniform isoperimetric constant. In particular, a countable group with non-vanishing first $\ell^2$-Betti number is uniformly non-amenable. We then define isoperimetric constants in the framework of measured equivalence relations. For an ergodic measured equivalence relation $R$ of type $\IIi$, the uniform isoperimetric constant $h(R)$ of $R$ is invariant under orbit equivalence and satisfies $$ 2\beta_1(R)\leq 2C(R)-2\leq h(R), $$ where $\beta_1(\R)$ is the first $\ell^2$-Betti number and $C(R)$ the cost of $R$ in the sense of Levitt (in particular $h(R)$ is a non-trivial invariant). In contrast with the group case, uniformly non-amenable measured equivalence relations of type $\IIi$ always contain non-amenable subtreeings. An ergodic version $h_e(\G)$ of the uniform isoperimetric constant $h(\G)$ is defined as the infimum over all essentially free ergodic and measure preserving actions $\alpha$ of $\G$ of the uniform isoperimetric constant $h(\R_\alpha)$ of the equivalence relation $R_\alpha$ associated to $\alpha$. By establishing a connection with the cost of measure-preserving equivalence relations, we prove that $h_e(\G)=0$ for any lattice $\G$ in a semi-simple Lie group of real rank at least 2 (while $h_e(\G)$ does not vanish in general)., Comment: 17 pp

We introduce concepts of intermediate rank for countable groups that "interpolate" between consecutive values of the classical (integer-valued) rank. Various classes of groups are proved to have intermediate rank behaviors. We are especially interested in interpolation between rank 1 and rank 2. For instance, we construct groups "of rank 7/4". Our setting is essentially that of non positively curved spaces, where concepts of intermediate rank include polynomial rank, local rank, and mesoscopic rank. The resulting framework has interesting connections to operator algebras. We prove property RD in many cases where intermediate rank occurs. This gives a new family of groups satisfying the Baum-Connes conjecture. We prove that the reduced $C^*$-algebras of groups of rank 7/4 have stable rank 1.

We are mainly interested here in Kazhdan's property T for measured equivalence relations. Among our main results are characterizations of strong ergodicity and Kazhdan's property in terms of the spectra of diffusion operators, associated to random walks and hilbertian representations of the underlying equivalence relation. The analog spectral characterization of property T for countable groups was proved recently by Gromov \cite{Gromov03} (and Ghys \cite{Ghys03}). Our proof put together the tools developed in the group case and further crucial technical steps from the study of amenable equivalence relations in \cite{CFW81}. As an application we show how \.Zuk's "$\lambda_1 >1/2$" criterion for property T can be adapted to measured equivalence relations., Comment: french, english summary

Recall Jones-Schmidt theorem that an ergodic measured equivalence relation is strongly ergodic if and only if it has no nontrivial amenable quotient. We give two new characterizations of strong ergodicity, in terms of metric-measured spaces. The first one identifies strong ergodicity with the concentration property as defined, in this (foliation) setting, by Gromov \cite{Gromov00_SQ}. The second one characterize the existence of nontrivial amenable quotients in terms of "F{\o}lner sequences" in graphings naturally associated to (the leaf space of) the equivalence relation. We also present a formalization of the concept of quasi-periodicity, based on (noncommutative) measure theory. The "singular measured spaces" appearing in the title refer to the leaf spaces of measured equivalence relations., Comment: french, english summary