Introduction to Sofic and Hyperlinear Groups and Connes' by Valerio Capraro, Martino Lupini, Vladimir Pestov

By Valerio Capraro, Martino Lupini, Vladimir Pestov

This monograph offers a few cornerstone ends up in the learn of sofic and hyperlinear teams and the heavily similar Connes' embedding conjecture. those notions, in addition to the proofs of many effects, are awarded within the framework of version idea for metric constructions. This standpoint, infrequently explicitly followed within the literature, clarifies the information therein, and offers extra instruments to assault open problems.

Sofic and hyperlinear teams are countable discrete teams that may be definitely approximated via finite symmetric teams and teams of unitary matrices. those deep and fruitful notions, brought by means of Gromov and Radulescu, respectively, within the past due Nineteen Nineties, prompted a magnificent volume of study within the final 15 years, touching numerous doubtless far away parts of arithmetic together with geometric staff conception, operator algebras, dynamical platforms, graph thought, and quantum details conception. numerous long-standing conjectures, nonetheless open for arbitrary teams, are actually settled for sofic or hyperlinear groups.

The presentation is self-contained and available to an individual with a graduate-level mathematical history. specifically, no particular wisdom of good judgment or version conception is needed. The monograph additionally includes many workouts, to assist familiarize the reader with the subjects present.

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Extra info for Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture

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In particular residually finite groups are sofic. More generally groups that are locally embeddable into amenable groups (LEA)—also called initially subamenable—are sofic. 3]). Therefore the previous discussion implies that free groups are sofic . Examples of hyperlinear and sofic groups that are not LEA have been recently constructed by Andreas Thom [143] and Yves de Cornulier [45], respectively. We will present these examples in Sect. 5. It should be now mentioned that it is to this day not know if there is any group which is not sofic, nor if there is any group which is not hyperlinear.

Suppose that € is a finitely generated group with finite generating set S D fg1 ; : : : ; gm g. n; "/-microstate for € is a map ˆ W € ! g1 ; : : : ; gn / D 1€ . n; "/-microstates for every n 2 N and " > 0. x/ defined by max fkxx 1k2 ; kx x 1k2 g . a/ D 0. C/ such that ka uk2 < ". x/ D 0 is close to an exact solution. C/ with an arbitrary tracial von Neumann algebra, can be established by means of the polar decomposition of an element inside a von Neumann algebra; see [14, Sect. 2]. 11 Using stability of the relation defining unitary elements, verify that the microstates formulation of hyperlinearity is equivalent to the original definition.

Equivalently the properties of being sofic or hyperlinear are elementary. Recall that the quantifiers in the logic for invariant metric groups are inf and sup. More precisely sup can be regarded as the universal quantifier, analogue to 8 in usual first order logic, while inf can be seen as the existential quantifier, which is denoted by 9 in the usual first order logic. A formula is therefore called universal if it only contains universal quantifiers, and no existential quantifiers. The notion of existential sentence is defined in the same way.

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