Property (T) for noncommutative universal lattices

Ershov, Mikhail; Jaikin–Zapirain, Andrei

doi:10.1007/s00222-009-0218-2

Cited by 63 publications

(105 citation statements)

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“…This result was improved by Ershov and Jaikin [13] to " ij Ä .n 1/ 1 . Moreover, Ershov and Jaikin [13], Theorem 5.9, also proved an analog of Theorem 1.2 in the case n D 3.…”

Section: Introductionmentioning

confidence: 94%

“…Here by "local representation theory" we mean studying the representations of (relatively) small subgroups in the group G. The second part of this method can be reduced to an optimization problem in some finite dimensional space, however in almost all cases the dimension is too big and this problem can not be approached directly. Instead, methods from linear algebra and graph theory are used (see [13], [14]). One unfortunate side effect is that the simple geometric idea behind this approach gets "hidden" in the computations.…”

Section: Introductionmentioning

confidence: 99%

“…Usually this is proved using the representation theory of the ambient Lie group [20], but this approach does not produce any bounds for the Kazhdan constants of this groups. In the last ten years several algebraic methods for proving property T have been developed [13], [18], [19], [23], [24]. One main advantage of these methods is that they provide explicit bounds for the Kazhdan constants of these groups, another is that these methods are applicable in a more general setting.…”

Section: Introductionmentioning

confidence: 99%

“…However, removing the condition p 5 (and replacing F p with Z) requires significant additional work, see [13] and [14]. Theorem 1.4 also can be generalized to show that many Steinberg and Kac-Moody groups have Kazhdan property T.…”

Section: Introductionmentioning

confidence: 99%

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Subspace arrangements and property T

Kassabov¹

2011

Groups Geom. Dyn.

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Abstract. We reformulate and extend the geometric method for proving the Kazhdan property T developed by Dymara and Januszkiewicz and used by Ershov and Jaikin. The main result says that a group G generated by finite subgroups G i has property T if the group generated by each pair of subgroups has property T and sufficiently large Kazhdan constant. Essentially, the same result was proven by Dymara and Januszkiewicz; however our bound for "sufficiently large" is significantly better.As an application of this result, we give exact bounds for the Kazhdan constants and the spectral gaps of the random walks on any finite Coxeter group with respect to the standard generating set, which generalizes a result of Bacher and de la Harpe.

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“…This result was improved by Ershov and Jaikin [13] to " ij Ä .n 1/ 1 . Moreover, Ershov and Jaikin [13], Theorem 5.9, also proved an analog of Theorem 1.2 in the case n D 3.…”

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Subspace arrangements and property T

Kassabov¹

2011

Groups Geom. Dyn.

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“…Instead the latter is proved in ref. 3 by adapting the proof of Theorem 1.5 and using conditions a-d more efficiently. In this paper we will introduce a generalized spectral criterion (Theorem 1.8), which can be used to prove not only Proposition 1.7, but its generalization Theorem 1.1.…”

Section: Proof Of Property (T) For El N (R) With R Arbitrarymentioning

confidence: 99%

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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Abelianization and fixed point properties of units in integral group rings

Bächle

Janssens

Jespers

et al. 2022

Mathematische Nachrichten

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Let 𝐺 be a finite group and  (ℤ𝐺) the unit group of the integral group ring ℤ𝐺. We prove a unit theorem, namely, a characterization of when  (ℤ𝐺) satisfies Kazhdan's property (T), both in terms of the finite group 𝐺 and in terms of the simple components of the semisimple algebra ℚ𝐺. Furthermore, it is shown that for  (ℤ𝐺), this property is equivalent to the weaker property FAb (i.e., every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property FA, denoted HFA. More precisely, it is described when all subgroups of finite index in  (ℤ𝐺) have both finite abelianization and are not a nontrivial amalgamated product. A crucial step for this is a reduction to arithmetic groups SL 𝑛 (), where  is an order in a finite-dimensional semisimple ℚ-algebra 𝐷, and finite groups 𝐺, which have the so-called cut property. For such groups 𝐺, we describe the simple epimorphic images of ℚ𝐺. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups E 𝑛 (𝐷) of SL 𝑛 (𝐷). These groups are well understood except in the degenerate case of lower rank, that is, for SL 2 () with  an order in a division algebra 𝐷 with a finite number of units. In this setting, we determine Serre's property FA for E 2 () and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its ℤ-rank.

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Property (T) for noncommutative universal lattices

Cited by 63 publications

References 16 publications

Subspace arrangements and property T

Subspace arrangements and property T

Groups graded by root systems and property ( T )

Abelianization and fixed point properties of units in integral group rings

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