# Projekt

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## Groupes sofiques: algèbre, analyse et dynamique

 Titel Englisch Sofic groups: algebra, analysis and dynamics Valette Alain 130435 Sinergia Institut de mathématiques Université de Neuchâtel Universität Neuenburg - NE Mathematik 01.05.2010 - 30.04.2013 841'014.00
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### Keywords (6)

Sofic groups; Hyperlinear groups; Surjunctive groups; Haagerup property (a-T-menability); C*-exactness; Uniform embeddings into Hilbert spaces

### Lay Summary (Englisch)

Lay summary
Sofic groups, by their nature, break frontiers between various areas of pure mathematics (geometric group theory, dynamical systems, operator algebras). This is a very promising class of groups whose recent genesis and modern understanding of geometric, asymptotic, and algebraic structures can generate new examples and theories in analytic properties of groups. It appears to be necessary to develop their study further and investigate connections with geometric group theory, topology, and dynamics.Analysis on discrete infinite groups is establishing itself as a new branch of group theory, with techniques borrowing from metric geometry, operator algebras, and harmonic analysis.Since 1980, Gromov is treating finitely generated groups as metric spaces, and a wealth of results by Gromov and followers show that a large part of the algebraic structure is captured by metric properties. In particular, Gromov introduced several asymptotic invariants of groups, i.e. quasi-isometry invariants, robust with respect to "local" perturbations and depending only on the "large-scale" geometry of the group. Many of these invariants are analytic in nature.The main goal of this project is to work out the precise relations between soficity and several group-theoretical properties which emerged recently, some analytic/algebraic (Haagerup property, C*-exactness, hyperlinearity,...), some metric (word hyperbolicity, Yu's property (A),...). Some of these properties imply positive results towards deep conjectures on group algebras, like the Baum-Connes conjecture, and our second objective is to contribute to some of these conjectures, by providing new classes of groups satisfying them.In this project we will focus on groups having geometric content, as a natural setup for doing analysis: hyperbolic groups, groups acting on trees, groups acting on Hilbert spaces, discrete subgroups of Lie groups.
 Direktlink auf Lay Summary Letzte Aktualisierung: 21.02.2013

### Verantw. Gesuchsteller/in und weitere Gesuchstellende

Name Institut
 Valette Alain Institut de Mathématiques Université de Neuchâtel
 Monod Nicolas Chaire de théorie ergodique et géométrique des groupes EPFL - SB - MATHGEOM - EGG
 Arjantseva Goulnara Fakultät für Mathematik Universität Wien

Name Institut

### Publikationen

Publikation
Capraro V Scarsini M (2013), Existence of equilibria in countable games: an algebraic approach, in Games and Economic Behavior, 79, 163-180.
V. Vapraro (2013), A discrete notion of continuity and applications, in Applied General Topology, 14(1), 61-72.
Brown N.P Capraro V. (2013), Groups associated to $II_1$-factors, in Journal of Functional Analysis , 264, 493-514.
N. Monod (2013), Groups of piecewise projective homeomorphisms, in PNAS, 110(12), 4524-4527.
S. Gal and J. Kedra (2012), A two-cocycle on the group of symplectic diffeomorphisms,, in Mathematische Zeitschrift, 271(3-4), 693-706.
Adam Timar (2012), Approximating Cayley graphs versus Cayley diagrams, in Combinatorics, probability and computing, 21, 635-641.
G. N. Arzhantseva E. Guentner and J. Spakula (2012), Coarse non-amenability and coarse embeddings, in Geometric and Functional Analysis, 22(1), 22-36.
G. N. Arzhantseva and E. Guentner (2012), Coarse non-amenability and covers with small eigenvalues, in Mathematische Annalen, 354(3), 863-870.
I. Benjamini O. Schramm and A. Timar (2012), On the separation profile of infinite graphs, in Groups, geometry and dynamics, 6(4), 639-658.
S. Burgdorf C. Scheiderer and M. Schweighofer (2012), Pure states, nonnegative polynomials and sums of squares, in Commentarii Mathematici Helvetici, 87(1), 113-140.
N. Monod (2011), A note on topological amenability, in IMRN, 17, 3872-3884.
S. Gal and J. Kedra (2011), On bivariant word metrics, in Journal of Topology and Analysis, 3(2), 161-175.
S. Gal and J. Kedra (2011), On distortion in groups of homeomorphisms, in Journal of Modern Dynamics, 5(3), 609-622.
N. Monod N. Ozawa (2010), The Dixmier problem, lamplighters and Burnside groups, in Journal of Functional Analysis, 258(1), 255-259.
Pierre-Nicolas Jolissaint Alain Valette, $\ell^p$-distortion and $p$-spectral gap of finite regular graphs, in Bull. London Math. Soc. .
L. Paunescu, A convex structure on sofic embeddings, in Ergodic Theory and Dynamical Systems.
A. Timar, A stationary random graph of no growth rate, in Annales Inst. Henri Poncaré.
V. Capraro, Amenability, locally finite spaces, and bi-Lipschitz embeddings, in Expo. Mathematicae.
V. Capraro S. Rossi, Banach spaces which embed into their dual, in Colloquium Math..
K. Juschenko N. Monod, Cantor systems, piecewise translations and simple amenable groups, in Annals of Mathematics, 178(2).
V. Capraro F. Radulescu, Cyclic Hilbert spaces and Connes' embedding problem, in Complex Analysis and Operator Theory.
P.-E. Caprace N. Monod, Fixed points and amenability in non-positive curvature, in Math. Annalen.
G. Arzhantseva J.-F. Lafont A. Minasyan, Isomorphism versus commensurability for a class of finitely presented groups,, in J. of Group Theory .
V. Capraro T. Fritz, On the axiomatization of convex subsets of a Banach space, in Proc. Amer. Math. Soc..
G. Elek N. Monod, On the topological full group of a minimal Cantor $\mathbf{Z}^2$-system, in Proc. Amer. Math. Soc..
S. Burgdorf K. Cafuta I. Klep J. Povh, The tracial moment problem and trace-optimization of polynomials, in Math. Programming, Ser. A.

### Verbundene Projekte

Nummer Titel Start Förderungsinstrument
 109130 Analyse géométrique sur les groupes et les variétés 01.10.2005 Projektförderung (Abt. I-III)
 163417 Géométrie et analyse sur les groupes discrets 01.10.2015 Projektförderung (Abt. I-III)
 126689 Analyse géométrique sur les groupes et les variétés 01.10.2009 Projektförderung (Abt. I-III)
 118014 Analyse géométrique sur les groupes et les variétés 01.10.2007 Projektförderung (Abt. I-III)

### Abstract

We plan to investigate two wide classes of groups which recently emerged at the crossroad of algebra, analysis and dynamical systems: sofic groups (i.e. subgroups of the metric ultra-product of symmetric groups) and hyperlinear groups (i.e. subgroups of the metric ultra-product of unitary groups). Equivalently, sofic (resp. hyperlinear) groups are those such that every finite set admit an almost homomorphism towards some symmetric (resp. unitary) group. These groups recently entered representation theory, K-theory, topological dynamics, theory of C*-algebras, and geometric group theory. Hyperlinear groups were motivated by Connes's embedding conjecture which asserts that every finite von Neumann algebra can be embedded into an ultra-product of finite-dimensional algebras, while sofic groups were introduced by Gromov in his study of symbolic algebraic varieties. In exploring these groups, we propose to develop new insight on various other classes of groups: a-T-menable groups (or groups with Haagerup property), C*-exact groups (appearing in remarkable work by Kirchberg), and groups coarsely embeddable into Hilbert space (introduced by Gromov in relation to the Novikov higher-signatures conjecture), etc. Our main objectives are: 1) To identify new broad classes of sofic groups. This will give both a new view on reputable theories (e.g. the theory of word hyperbolic groups) and alternative approaches to infinite groups. 2) To perform an overall study of analytic nature of infinite "monster" groups through their geometric and asymptotic properties. 3) To compare soficity and hyperlinearity to other analytical properties of discrete groups (a-T-menability, exactness, uniform embeddability of groups into Hilbert spaces) and study further permanence properties of the corresponding classes of groups. This will considerably extend the class of groups for which celebrated conjectures (due to Baum and Connes, Farrell and Jones, Kaplansky, Bass, etc.) are satisfied. In particular, we hope to solve the ambitious questions: Is every countable group sofic? Hyperlinear?'' A negative answer would provide a new generation exotic infinite group that is required for counterexamples. A positive answer would refute, in particular, the famous meta-theorem of Gromov claiming that a proposition which holds for all countable discrete groups is either trivial or false.
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