LEADER 02818nam a2200361 i 4500 001 991003225059707536 008 160728s2015 sz b 001 0 eng d 020 $a9783319193328 035 $ab14299811-39ule_inst 040 $aBibl. Dip.le Aggr. Matematica e Fisica - Sez. Matematica$beng 082 04$a512 084 $aAMS 20F65 084 $aAMS 03C20 084 $aAMS 03C65 084 $aAMS 37A15 084 $aLC QA174.2.C37 100 1 $aCapraro, Valerio$0717976 245 10$aIntroduction to sofic and hyperlinear groups and connes' embedding conjecture /$cValerio Capraro, Martino Lupini ; with an appendix by Vladimir Pestov 260 $aCham [Switzerland] :$bSpringer,$cc2015 300 $aviii, 151 p. ;$c24 cm 440 0$aLecture notes in mathematics,$x0075-8434 ;$v2136 504 $aIncludes bibliographical references and index 505 0 $aIntroduction ; Sofic and hyperlinear groups ; Connes' embedding conjecture ; Conclusions 520 $aThis monograph presents some cornerstone results in the study of sofic and hyperlinear groups and the closely related Connes' embedding conjecture. These notions, as well as the proofs of many results, are presented in the framework of model theory for metric structures. This point of view, rarely explicitly adopted in the literature, clarifies the ideas therein, and provides additional tools to attack open problems. Sofic and hyperlinear groups are countable discrete groups that can be suitably approximated by finite symmetric groups and groups of unitary matrices. These deep and fruitful notions, introduced by Gromov and Radulescu, respectively, in the late 1990s, stimulated an impressive amount of research in the last 15 years, touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and quantum information theory. Several long-standing conjectures, still open for arbitrary groups, are now settled for sofic or hyperlinear groups. The presentation is self-contained and accessible to anyone with a graduate-level mathematical background. In particular, no specific knowledge of logic or model theory is required. The monograph also contains many exercises, to help familiarize the reader with the topics present 650 0$aGroup theory 650 0$aOperator theory 700 1 $aLupini, Martino$eauthor$4http://id.loc.gov/vocabulary/relators/aut$0721405 907 $a.b14299811$b22-11-16$c28-07-16 912 $a991003225059707536 945 $aLE013 20F CAP11 (2015)$g1$i2013000293837$lle013$op$pE36.39$q-$rl$s- $t0$u1$v0$w1$x0$y.i15789056$z22-11-16 996 $aIntroduction to sofic and hyperlinear groups and Connes' embedding conjecture$91413246 997 $aUNISALENTO 998 $ale013$b28-07-16$cm$da $e-$feng$gsz $h0$i0