Symmetric groups and expander graphs

Kassabov, Martin

doi:10.1007/s00222-007-0065-y

Cited by 90 publications

(61 citation statements)

References 32 publications

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“…As we will see in Section 3, Theorem 1.1 is an easy consequence of the following theorem, together with recent results of Kassabov [5] and Ellis-Hachtman-SchneiderThomas [3].…”

Section: Centralizers and Expander Familiesmentioning

confidence: 58%

“…In Section 2, we will use a basic property of expander graphs to show that certain ultraproducts D G n of finite groups can be realized as centralizers of finitely generated subgroups in suitably chosen universal sofic groups. Then in Section 3, using recent results of Kassabov [5] and EllisHachtman-Schneider-Thomas [3], we will complete the proof of Theorem 1.1.…”

Section: Introductionmentioning

confidence: 96%

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On the number of universal sofic groups

Thomas

2010

Proc. Amer. Math. Soc.

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“…As we will see in Section 3, Theorem 1.1 is an easy consequence of the following theorem, together with recent results of Kassabov [5] and Ellis-Hachtman-SchneiderThomas [3].…”

Section: Centralizers and Expander Familiesmentioning

confidence: 58%

Section: Introductionmentioning

confidence: 96%

On the number of universal sofic groups

Thomas

2010

Proc. Amer. Math. Soc.

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“…In another breakthrough [Kas05b], Kassabov showed that the family of alternating and symmetric groups A n and S n can also be made into a family of boundeddegree expanders. His construction combines ingenious combinatorial arguments with estimates on characters of the symmetric group due to Roichman [Roi96].…”

Section: Cayley Expander Graphsmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,502

1,355

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A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields.But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are "just" one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early '70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even be able to mention. This universality of expanders is becoming more evident as more connections are discovered. It transpires that expansion is a fundamental mathematical concept, well deserving to be thoroughly investigated on its own.In hindsight, one reason that expanders are so ubiquitous is that their very definition can be given in at least three languages: combinatorial/geometric, probabilistic and algebraic. Combinatorially, expanders are graphs which are highly connected; to disconnect a large part of the graph, one has to sever many edges. Equivalently, using the geometric notion of isoperimetry, every s...

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“…One of the main theorems in this area asserts that all (nonabelian) finite simple groups form a family of expanders. The proof of this result is spread over several papers (16)(17)(18).…”

Section: Application To Expandersmentioning

confidence: 97%

Groups graded by root systems and property ( T )

Ershov¹,

Jaikin–Zapirain²,

Kassabov³

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

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We establish property (T) for a large class of groups graded by root systems, including elementary Chevalley groups and Steinberg groups of rank at least 2 over finitely generated commutative rings with 1. We also construct a group with property (T) which surjects onto all finite simple groups of Lie type and rank at least two.property (T) | root system gradings | Steinberg groups | Chevalley groups G roups graded by root systems can be thought of as natural generalizations of Steinberg and Chevalley groups over rings. In recent preprints (1, 2), the authors of this paper determined a sufficient condition which almost implies property ðTÞ for a group graded by a root system (see Theorem 1.1 below) and used this result to establish property ðTÞ for Steinberg and Chevalley groups corresponding to reduced irreducible root systems of rank at least 2. The goal of this paper is to give an accessible exposition of those results and describe the main ideas used in their proofs.As will be shown in ref. 1, a substantial part of the general theory of groups graded by root systems can be developed using the term "root system" in a very broad sense. However, the majority of interesting examples (known to the authors) come from classical root systems (that is, finite crystallographic root systems), so in this paper we will only consider classical root systems (and refer to them simply as root systems).The basic idea behind the definition of a group graded by a root system Φ is that it should be generated by a family of subgroups indexed by Φ, which satisfies commutation relations similar to those between root subgroups of Chevalley and Steinberg groups.Definition: Let G be a group, Φ a root system, and fX α g α∈Φ a family of subgroups of G. We will say that the groups fX α g α∈Φ form a Φ-grading if for any α; β ∈ Φ with β ∉ Rα, the following inclusion holds:If in addition G is generated by the subgroups fX α g, we will say that G is graded by Φ and that fX α g α∈Φ is a Φ-grading of G. The groups X α themselves will be referred to as root subgroups.Clearly, the above definition is too general to yield any interesting structural results, and we are looking for the more restrictive notion of a strong grading. A sufficient condition for a Φ-grading to be strong is that the inclusion in [1.1] is an equality; however, requiring equality in general is too restrictive, as, for instance, it fails for Chevalley groups of type B n over rings where 2 is not invertible. To formulate the definition of a strong grading in the general case, we need some additional terminology.Let Φ be a root system in a Euclidean space V with inner product ð·;·Þ. A subset B of Φ will be called Borel if B is the set of positive roots with respect to some system of simple roots Π of Φ. Equivalently, B is Borel if there exists v ∈ V , which is not orthogonal to any root in Φ, such that B = fγ ∈ Φ : ðγ; vÞ > 0g.If B is a Borel subset and Π is the associated set of simple roots, the boundary of B, denoted by ∂B, is the set of roots in B which are multiples of roo...

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Symmetric groups and expander graphs

Cited by 90 publications

References 32 publications

On the number of universal sofic groups

On the number of universal sofic groups

Expander graphs and their applications

Groups graded by root systems and property ( T )

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