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020 _a9783319193335
_9978-3-319-19333-5
024 7 _a10.1007/978-3-319-19333-5
_2doi
035 _ato000560061
040 _aSpringer
_cSpringer
_dRU-ToGU
050 4 _aQA174-183
072 7 _aPBG
_2bicssc
072 7 _aMAT002010
_2bisacsh
082 0 4 _a512.2
_223
100 1 _aCapraro, Valerio.
_eauthor.
_9466843
245 1 0 _aIntroduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture
_helectronic resource
_cby Valerio Capraro, Martino Lupini.
250 _a1st ed. 2015.
260 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2015.
300 _aVIII, 151 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
490 1 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v2136
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 _amathematics.
_9566183
650 0 _aGroup theory.
_9303360
650 0 _aOperator theory.
_9566350
650 1 4 _aMathematics.
_9566184
650 2 4 _aGroup Theory and Generalizations.
_9303362
650 2 4 _aOperator Theory.
_9566351
700 1 _aLupini, Martino.
_eauthor.
_9466844
710 2 _aSpringerLink (Online service)
_9143950
773 0 _tSpringer eBooks
830 0 _aLecture Notes in Mathematics,
_9315276
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-319-19333-5
912 _aZDB-2-SMA
912 _aZDB-2-LNM
999 _c414792