Sofic groups, by their nature, bridge divides 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 already generates new examples and investigations 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. [LaWiNe poster]
The Folner function for an amenable group G with a finite generating set S is a quantitative measure of "how amenable" the group is; the nth value of the Folner function is the minimum cardinality of a 1/n Folner set. While Thompson's group is not known to be amenable, there is a constant C such that the minimum cardinality of a C^{-n} Folner set is at least tower(n). (Here tower(0) = 1 and tower(n+1) = 2^tower(n).) This growth is a consequence of a qualitative property — "monotonicity" — which invariant measures for F must exhibit.
I will give a formal sense in which Thompson's group is in fact a Ramsey-theoretic problem. The monotonicity phenomenon and growth rates discussed in the first talk fit naturally with this viewpoint and in turn suggest a natural and novel line of attack on the problem. I will discuss some partial successes along these lines and also some pitfalls.
We consider the notion of a surjunctive map (and, more generally, of a surjunctive (concrete) category) and the related notion of a surjunctive group (due to W. Gottschalk) in the dynamical setting of cellular automata on groups. We show that residually finite groups are surjunctive (for a large class of categories including those of finite sets, of finite-dimensional vector spaces, and of affine algebraic sets over uncountable algebraically closed fields).
We show that amenable groups and, more generally, sofic groups are surjunctive (for a large class of categories including those of finite sets and of finite-dimensional vector spaces). We finally discuss the notion of a surjunctive subshift and the related problem of the density of periodic configurations in general subshifts.
A few years ago Bowen introduced a notion of entropy for measure-preserving actions of sofic groups and used it to obtain a far-reaching extension of the Ornstein-Weiss classification of Bernoulli actions over amenable groups. Subsequently Hanfeng Li and I developed a more general operator-algebraic approach to sofic entropy and established a variational principle in this context. I will show that these two perspectives can be reconciled to produce a definition with the novelty that it does not depend on generators, like the standard formulation of classical measure entropy due to Sinai. This leads to a streamlined analysis of the entropy of Bernoulli actions over sofic groups, and in particular enables one to show that such actions have completely positive entropy.
I will discuss some recent work with Hanfeng Li in which we initiate a study of combinatorial independence in the context of sofic entropy. The project involves, among other things, ergodic theory on ultraproducts and applications to the Fuglede-Kadison determinant in group von Neumann algebras.
Homoclinic points describe the asymptotic behavior of group actions on spaces and play an important role in general theory of dynamical systems. In 1999, Doug Lind and Klaus Schmidt established relations between homoclinic points and entropy properties for expansive algebraic actions of $Z^d$. Their proof depends heavily on the commutative factorial Noetherian ring structure of the integral group ring of $Z^d$. In a joint work with Hanfeng Li, we extend their results to expansive algebraic actions of polycyclic-by-finite groups. We use three ingredients to do this: characterizations of expansive algebraic actions, local entropy theory for actions of countable amenable groups on compact groups, and comparison between entropies of dual algebraic actions. Applying our results to the field of von Neumann algebras, we get a positive answer to a question of Deninger about the Fuglede-Kadison determinant to the case the group is amenable. We also prove that for an amenable group, an element in the integral group ring is a non-zero divisor if and only if the entropy of the corresponding principal algebraic action is finite.
In 2008, in a remarkable breakthrough, via modeling the dynamics of a measurable partition of probability space by means of partitions of a finite space, Lewis Bowen showed how to define entropy for measure-preserving actions of countable sofic groups. Later, using ideas in operator algebras, David Kerr and Hanfeng Li, developed a more general approach for both measure and topological sofic entropies and established the variational principle for this context. In this talk, applying Kerr and Li's method, I will define the topological pressure for actions of sofic groups and establish the variational principle for topological pressure in the sofic case.
In recent work Lewis Bowen developed notions of entropy for probability measure preserving actions of finitely generated free groups, known as f-invariant entropy, and actions of sofic groups, known as sofic entropy. These entropies are closely related to one another and have strong similarities to the classical Kolmogorov—Sinai entropy of actions of amenable groups. In this talk I will review what is known about f-invariant entropy and I will discuss what it means for an action of a finitely generated free group to have defined and finite f-invariant entropy.
Burnside's Problem and the von Neumann Conjecture are classical problems from group theory which were long ago answered in the negative. In 1999, Kevin Whyte defined geometric analogs of these problems and proved the Geometric von Neumann Conjecture. In this talk, I will present a solution to the Geometric Burnside's Problem. I will also present a strengthening of Whyte's result and draw conclusions about the existence of regular spanning trees of Cayley graphs.
The L^{p} compression of an infinite metric space measures how well this space embeds into L^{p}; similarly, L^{p}-distortion of a finite metric space measures how well it imbeds into L^{p} . If X is the disjoint union of a family (X_{n})_{{n>0}} of finite metric spaces, a lemma by T. Austin shows that asymptotic lower bounds on the distortion of the X_{n}'s, imply upper bounds on the compression of X. When X_{n} is a finite k-regular connected graph, we provide a lower bound on its distortion, involving the first non-zero eigenvalue of the p-laplacian. This inequality provides a new proof of the Linial-Magen result that families of expanders have "the worst possible" distortion, i.e. it is logarithmic in the number of vertices. This is joint work with P.-N. Jolissaint.
Symbolic dynamics on Z^{d} is concerned with the dynamics of the action of Z^{d} by translation on the space of symbolic configurations (colorings of Z^{d} by a finite alphabet), and the closed sets invariant for this action, called subshifts. Subshifts of finite type, that can be defined by local constraints, are of particular interest since they can model many real-life phenomena. In this talk I will present some recent results about the complexity of these multidimensionnal subshifts of finite type.
In this second talk I will present some problems of symbolic dynamics on finitely presented groups : the domino problem and its relation to the word problem, the existence of aperiodic subshift of finite type, the soficity of the one-or-less subshift. The underlying motivation is to classify finitely presented groups depending on the properties they induce on susbhifts.
I will introduce logic for metric structures, the notion of ultraproduct in this setting and their relations with the subject of sofic and hyperlinear groups. No specific background in logic will be assumed.
I will present the main ideas and techniques of the proof, obtained by means of logic for metric structures, that there are consistently power of the continuum many pairwise non isomorphic universal sofic and hyperlinear groups. This answers a question from 2010 of Simon Thomas, who proved this statement for universal sofic groups with an ad hoc algebraic argument, and asked if the same was true for universal hyperlinear groups.
Murray von-Neumann dimension for representations of a discrete group contained in a multiple of the left-regular representations is well known and was developed by von Neumann in 1936. An immediate corollary of its basic properties is that the representations l^{2}(G)^{⊕n} are pairwise non-isomorphic for different values of n if G is discrete. In 1999, Gromov asked whether this generalizes to l^{p}(G)^{⊕n} for 1 ≤ p < ∞, and proposed a method for proving this when G is amenable. A. Gournay carried out his method in 2010, and proved that l^{p}(G)^{⊕n} are pairwise non-isomorphic for different values of n, by extending the notion of Murray-von Neumann dimension to closed invariant subspaces of l^{p}(G)^{⊕n}. The main idea for this extension is to treat the representations l^{p}(G)^{⊕n} as analogous to Bernoulli shifts, and Murray von-Neumann dimension as like entropy. Following ideas of L. Bowen, A. Gournay, D. Kerr, H. Li, and D. Voiculescu we define a notion of l^{p} dimension for uniformly bounded representations of a countable discrete sofic group on a separable Banach space, further this notion extends to a version of S^{p} (for Schatten p class operators) for hyperlinear groups. In particular, we show that l^{p}(G)^{⊕n} are non-isomorphic as a representation of G for different values of n if G is hyperlinear. We also define l^{p}-Betti numbers, and compute l^{p}-Betti numbers of F_{n}. Lastly, we compute dimension for actions of a hyperlinear group on noncommutative L^{p} spaces.
Hyperfiniteness or amenability of measurable equivalence relations and group actions has been studied for almost fifty years. Recently, unexpected applications of hyperfiniteness were found in computer science in the context of testability of graph properties. Motivated by the work of Lovasz and Szegedy on graph limits, in this paper we propose a unified approach to hyperfiniteness. We establish some new results and give new proofs of the theorems of Schramm, Lovasz and Newman-Sohler.
We will discuss group theoretic properties that preserve surjunctivity. In particular, we will discuss semidirect products.
A finitely generated group is sofic if its Cayley diagram (the Cayley graph edge-labeled with the generators) can be approximated by finite diagrams. Here, approximation is meant in the so-called Benjamini-Schramm metric. We get a weaker notion of soficity if we forget the edge-labels (that is, we want to approximate the Cayley graph of our group by finite graphs). In this talk I present an even weaker form of approximation, related to the moments of the spectral measure of the graphs. I will explain how this might help us deciding whether every group is sofic.
TBA
We prove a fixed point theorem for a family of Banach spaces including notably L^{1} and its non-commutative analogues. Several applications are given, e.g. the optimal solution to the "derivation problem" studied since the 1960s.
In this talk, I will shortly describe the von Neumann dimension (or G-dimension) for a discrete and finitely generated group G. In a particular case, this is a (non-negative) real number which can be associated to closed G-invariant subspaces of l^{2}(G) which behaves like one expects a dimension to behave. When the group is amenable, it is possible to define such a dimension from a more naive point of view, in particular, one which does not hinge on the Hilbertian structure. This allows to define a "dimension" for subspaces of other classical Banach spaces, e.g. l^{p}(G).
The tracial moment problem is an extension of the classical moment problem to tracial linear forms on the ring of polynomials in noncommuting variables. It deals with the question which linear functionals are given by tracial moments of a probability measure on symmetric matrices. This problem is motivated by Connes' embedding problem and its relation to noncommutative Real Algebra shown by Klep and Schweighofer. After a short introduction partial solutions of this problem generalizing results of Haviland and of Curto and Fialkow for the classical moment problem will be presented.
Classically entropy was defined for actions of countable amenable groups in both topological and measurable setting. In 2008 Lewis Bowen defined entropy for measure-preserving actions of a countable sofic group G on probability measure spaces when there exists a countable generating partition with finite Shannon entropy. I will discuss how to define entropy for any continuous action of G on compact metrizable spaces and measure-preserving actions of G on standard probability measure spaces. This is joint work with David Kerr.
The local entropy theory for actions of countable amenable groups was initiated by Blanchard twenty years ago. An important piece of this theory is the product formula for entropy tuples. I will discuss how to extend the local entropy theory to actions of countable sofic groups, and how the product formula for entropy tuples in sofic setting relates to the ergodicity of the commutant of the sofic group acting on the Loeb measure space. This is joint work with David Kerr.
TBA
We will consider convex sets of matrices composed of second-order mixed moments of n unitaries with respect to finite traces. These sets are of interest in connection with Connes' embedding problem. We overview the basic facts about Connes' embedding problem. In particular, we will discuss a theorem of E. Kirchberg which was the main motivation to study matrices of second-order moments. We present some properties of these sets and descriptions in case of small n. We also discuss a connection with the sets of correlation matrices and give some related examples. We will present a modification of I. Klep and M. Schweighofer algebraic reformulation of Connes' embedding problem by considering *-algebra of the countably generated free group. This allows to consider only quadratic polynomials in unitary generators instead of arbitrary polynomials in self-adjoint generators.
In 1976, Alain Connes posed a problem in the theory of von Neumann algebras which over the years turned out to be equivalent or closely related to numerous important problems in operator algebra, free probability and quantum computing (such as Kirchberg's QWEP conjecture, Voiculescu's free entropy or Tsirnelson's problem). Despite the increasing interest in this conjecture, it resisted so far all attempts of proving or disproving it. The original version of the problem asks whether each separable $II_1$-factor can be embedded into any countable free ultraproduct of the hyperfinite $II_1$-factor. It can also be formulated as a question on approximation tracial $II_1$-moments by tracial matrix moments. In this talk, we present Connes' embedding problem and give yet another reformulation due to Klep and myself in the flavor of real algebraic geometry. This reformulation is purely algebraic (in particular, von Neumann algebras do no longer appear in it) and restates Connes' problem in terms of sums of squares certificates for trace positivity of polynomials. Interestingly, both the commutative and non-commutative variant of Connes' embedding problem are both known to be true by theorems of Putinar from 1993 and of Helton and McCullough from 2004.
This talk is partially motivated by the previous talk on Connes' embedding problem but does not depend on it. We investigate Connes' embedding problem from the viewpoint of its purely algebraic reformulation in terms of trace positivity of polynomials. The object of study are thus polynomials in non-commuting variables which satisfy certain trace positivity conditions, i.e., have nonnegative trace whenever one substitutes the variables by certain symmetric matrices of the same size. The question is when and how such polynomials can be represented in a way that makes their trace positivity property obvious. Ideally such a representation would for example be a sum of commutators and hermitian squares of polynomials. We present amongst others recent theorems obtained in the thesis of Sabine Burgdorf on this subject.
The notion of sofic group was introduced by Gromov in 1999. In 2009 Elek and Lippner introduced sofic equivalence relations. We shall present these notions in an operator algebraic context starting from Connes' Embedding Problem, a central open problem in the theory of operator algebra. We shall prove that amenable groups have essentially one sofic representation up to conjugacy. For this result we shall use the powerful theorems in the theory of von Neumann algebras about amenable groups. Using this result we can prove that an amalgamated product of sofic groups over an amenable group is again sofic. We shall also need to show that Bernoulli shifts of sofic groups are sofic, a result from the original article of Elek and Lippner. This results are contained in the article arXiv:1002.0605 and were obtained independently by Elek and Szabo using different arguments, arXiv:1010.3424.
A sofic approximation is a family of finite models that approximate better and better a given object (typically a countable group, but this can be applied more generally, e.g. to group actions) to within some prescribed tolerance. I will explain how to obtain results about orbit equivalence for measure preserving actions of countable groups by "counting" the number of sofic approximations for these actions. This is joint work with Ken Dykema and David Kerr.
Suppose G is an infinite group and H is a finite subgroup of
G. Consider an element T of the integral group ring of G, whose
support is contained in H. If we consider T as an operator on the
Hilbert space l^{2}(G) then it is very easy to understand the spectrum
of T, since it is the same as understanding spectrum of the operator T
acting on the finite-dimensional Hilbert space l^{2}(H).
In the talk I will consider a situation when support of T generates
the whole group G and so there is no possibility of passing to a
smaller subgroup in order to simplify computations. However, in
favourable cases integral group ring of G embeds into the groupoid
ring of a certain measured groupoid, and in this groupoid it is
actually possible to pass to a subgroupoid whose groupoid ring still
contains T, and in which the range and source maps have finite fibers
(which is a groupoid analogue of being a finite group). This still
allows to simplify computation of the spectrum of T considerably.
If need arises I'll explain what measured groupoids are. The most
important example relevant to the presented techniques is the relation
groupoid of a measurable relation.
As an example I will present some computations for the lamplighter
group G relevant for the Atiyah problem for G (which I'll also explain
- roughly, Atiyah problem asks about possible measures of atoms in
spectra of elements of the group ring of G). If time permits I'll also
explain how a Turing machine gives rise to a group G and T in the
group ring of G, and how - using the computational technique
described - properties of the Turing machine can be related to
spectral properties of T.
TBA
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I will try to present the content of the eponymous article by Gromov in which he introduces sofic groups. The basis is a theorem of Ax: every regular algebraic map from a complex algebraic variety S to itself is surjunctive (i.e. is not a strict embedding). The proof of this theorem hinges on the (extended) Lefschetz principle, which states roughly that a first-order statement is true on an algebraically closed field of characteristic zero if and only if it holds for all algebraically closed field of characteristic p. This approximation of models is then used to extend this result. If one takes the infinite power of the variety S indexed by a group G, then a similar result holds for G-equivariant pro-regular maps provided the group G itself can be approximated by "nice" groups (e.g. finite groups approximate residually finite groups). It turns out that amenable groups are also "nice", and sofic groups are then defined as those groups that can be approximated by amenable groups.
In a joint work with L. Paunescu, we have used the notion of product of ultrafilters in order to get easier and more general version of Radulescu's and Elek-Szabo's theorems about hyperlinear groups. Moreover we are able to construct two non-trivial examples of hyperlinear group: the free group on uncountably many generators and the unit sphere S^{1}.