Following the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\ℍn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign.As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles.In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

Johnson's characterization of amenable groups states that a discrete group Γ is amenable if and only if H n≥1 b (Γ; V ) = 0 for all dual normed R[Γ]-modules V . In this paper, we extend the previous result to homomorphisms by proving the converse of the Mapping Theorem: a surjective homomorphism ϕ : Γ → K has amenable kernel H if and only if the induced inflation mapIn addition, we obtain an analogous characterization for the (smaller) class of surjective homomorphisms ϕ : Γ → K with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all normed R[Γ]-modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic homomorphisms ϕ : Γ → K, for which the restriction maps in bounded cohomology HWe then extend the first and third results to spaces and obtain characterizations of amenable maps and boundedly acyclic maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.

We provide new computations in bounded cohomology: A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated non-amenable boundedly acyclic groups and that there exists a finitely presented boundedly acyclic group that is universal in the sense that it contains all finitely presented groups.On the other hand, we construct a continuum of finitely generated groups, whose bounded cohomology has uncountable dimension in all degrees greater than or equal to 2. Countable non-amenable groups with these two extreme properties were previously known to exist, but these constitute the first finitely generated examples.Finally, we show that various algorithmic problems on bounded cohomology are undecidable.

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume and bounded cohomology.In order to study the main properties of simplicial volume, Gromov himself initiated the dual theory of bounded cohomology, which then developed into a very active and independent research field. Gromov's theory of bounded cohomology of topological spaces was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets.The first aim of this paper is to lay the foundation of the theory of multicomplexes. After setting the main definitions, we construct the singular multicomplex K(X) associated to a topological space X, and we prove that the geometric realization of K(X) is homotopy equivalent to X for every CW complex X. Following Gromov, we introduce the notion of completeness, which, roughly speaking, translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes, and we show that K(X) is complete for every CW complex X.In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov's Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov's Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity.The third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide complete proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the 1 -invisibility of closed manifolds in terms of amenable covers. As an application, we give the first complete proof of the vanishing of the simplicial volume of the product of three open manifolds. Contents 6.3. Proof of the Vanishing Theorem II 6.4. Proof of the Vanishing Theorem I Part 3. The simplicial volume of open manifolds Chapter 7. The Finiteness and the Vanishing Theorems 7.1. The simplicial volume of open manifolds 7.2. The Vanishing and the Finiteness Theorems 7.3.1 -homology and invisibility Chapter 8. Diffusion of chains 8.1. Diffusion operators 8.2. Locally finite actions and diffusion 8.3. A toy example Chapter 9. Admissible submulticomplexes of K(X) 9.1. (Strongly) admissible simplices and admissible maps 9.2. Admissible multicomplexes 9.3. Group actions on the admissible multicomplex 9.4. Amenable subgroups of Aut AD (AD L (X)) Chapter 10. The proofs of the Vanishing and the Finiteness Theorems 10.1. An important locally finite action 10.2. Locally finite functions versus locally finite ch...

The problem of a correct fall risk assessment is becoming more and more critical with the ageing of the population. In spite of the available approaches allowing a quantitative analysis of the human movement control system's performance, the clinical assessment and diagnostic approach to fall risk assessment still relies mostly on non-quantitative exams, such as clinical scales. This work documents our current effort to develop a novel method to assess balance control abilities through a system implementing an automatic evaluation of exercises drawn from balance assessment scales. Our aim is to overcome the classical limits characterizing these scales i.e. limited granularity and inter-/intra-examiner reliability, to obtain objective scores and more detailed information allowing to predict fall risk. We used Microsoft Kinect to record subjects' movements while performing challenging exercises drawn from clinical balance scales. We then computed a set of parameters quantifying the execution of the exercises and fed them to a supervised classifier to perform a classification based on the clinical score. We obtained a good accuracy (~82%) and especially a high sensitivity (~83%).