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| In [[mathematics]], a [[locally compact]] [[topological group]] ''G'' has '''property (T)''' if the [[trivial representation]] is an [[isolated point]] in its [[unitary dual]] equipped with the [[Spectrum of a C*-algebra|Fell topology]]. Informally, this means that if ''G'' acts [[unitary representation|unitarily]] on a [[Hilbert space]] and has "almost invariant vectors", then it has a nonzero [[invariant vector]]. The formal definition, introduced by [[David Kazhdan]] ([[#CITEREFKazhdan1967|1967]]), gives this a precise, quantitative meaning.
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| Although originally defined in terms of [[irreducible representation]]s, property (T) can often be checked even when there is little or no explicit knowledge of the unitary dual. Property (T) has important applications to [[group representation theory]], [[Grigory Margulis|lattices in algebraic groups over local fields]], [[ergodic theory]], [[geometric group theory]], [[Expander graph|expanders]], [[operator algebras]] and the [[expanding graph|theory of networks]].
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| ==Definitions==
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| Let ''G'' be a σ-compact, locally compact [[topological group]] and π : ''G'' → ''U''(''H'') a [[unitary representation]] of ''G'' on a (complex) Hilbert space ''H''. If ε > 0 and ''K'' is a compact subset of ''G'', then a unit vector ξ in ''H'' is called an '''(ε, ''K'')-invariant vector''' if
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| :<math> \forall g \in K \ : \ \left \|\pi(g) \xi - \xi \right \| < \varepsilon.</math>
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| The following conditions on ''G'' are all equivalent to ''G'' having '''property (T)''' of [[David Kazhdan|Kazhdan]], and any of them can be used as the definition of property (T).
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| (1) The [[trivial representation]] is an [[isolated point]] of the [[unitary dual]] of ''G'' with [[Fell topology]].
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| (2) Any sequence of [[continuous function|continuous]] [[positive definite function on a group|positive definite functions]] on ''G'' converging to 1 [[uniform convergence|uniformly]] on [[compact subset]]s, converges to 1 uniformly on ''G''.
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| (3) Every [[unitary representation]] of ''G'' that has an (ε, ''K'')-invariant unit vector for any ε > 0 and any compact subset ''K'', has a non-zero invariant vector.
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| (4) There exists an ε > 0 and a compact subset ''K'' of ''G'' such that every unitary representation of ''G'' that has an (ε, ''K'')-invariant unit vector, has a nonzero invariant vector.
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| (5) Every continuous [[affine transformation|affine]] [[isometry|isometric]] [[group action|action]] of ''G'' on a ''real'' [[Hilbert space]] has a fixed point ('''property (FH)''').
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| If ''H'' is a [[closed subgroup]] of ''G'', the pair (''G'',''H'') is said to have '''relative property (T)''' of [[Gregory Margulis|Margulis]] if there exists an ε > 0 and a compact subset ''K'' of ''G'' such that whenever a unitary representation of ''G'' has an (ε, ''K'')-invariant unit vector, then it has a non-zero vector fixed by ''H''.
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| == Discussion ==
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| Clearly, definition (4) implies definition (3). Let us show the converse, ''assuming local compactness''. So let ''G'' be a locally compact group satisfying (3). By Theorem 1.3.1 of Bekka et al., ''G'' is compactly generated. Therefore, Remark 1.1.2(v) of Bekka et al. tells us the following. If we take ''K'' to be a compact generating set of ''G'', and let ε be any positive real number, then a unitary representation of ''G'' having an (ε, ''K'')-invariant unit vector has (ε', ''K'' ')-invariant unit vectors for every ε' > 0 and ''K'' ' compact. Therefore, by (3), such a representation of ''G'' will have a nonzero invariant vector, establishing (4).
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| The equivalence of (4) and (5) (Property (FH)) is the Delorme-Guichardet Theorem. The fact that (5) implies (4) requires us to assume that ''G'' is σ-compact (and locally compact) (Bekka et al., Theorem 2.12.4).
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| == General properties ==
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| * Property (T) is preserved under quotients: if ''G'' has property (T) and ''H'' is a [[quotient group]] of ''G'' then ''H'' has property (T). Equivalently, if a homomorphic image of a group ''G'' does ''not'' have property (T) then ''G'' itself does not have property (T).
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| * If ''G'' has property (T) then ''G''/[''G'', ''G''] is compact.
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| * Any countable discrete group with property (T) is finitely generated.
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| * An [[amenable group]] which has property (T) is necessarily [[compact group|compact]]. Amenability and property (T) are in a rough sense opposite: they make almost invariant vectors easy or hard to find.
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| * '''Kazhdan's theorem''': If Γ is a [[lattice (discrete subgroup)|lattice]] in a Lie group ''G'' then Γ has property (T) if and only if ''G'' has property (T). Thus for ''n'' ≥ 3, the special linear group SL(''n'', '''Z''') has property (T).
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| ==Examples==
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| * [[Compact topological group]]s have property (T). In particular, the [[circle group]], the additive group '''Z'''<sub>''p''</sub> of ''p''-adic integers, compact [[special unitary group]]s SU(''n'') and all finite groups have property (T).
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| * [[Simple Lie group|Simple]] real [[Lie group]]s of real [[rank of a Lie group|rank]] at least two have property (T). This family of groups includes the [[special linear group]]s SL(''n'', '''R''') for ''n'' ≥ 3 and the special [[orthogonal group]]s SO(''p'',''q'') for ''p'' > ''q'' ≥ 2 and SO(''p'',''p'') for ''p'' ≥ 3. More generally, this holds for simple [[algebraic group]]s of rank at least two over a [[local field]].
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| * The pairs ('''R'''<sup>''n''</sup> ⋊ SL(''n'', '''R'''), '''R'''<sup>''n''</sup>) and ('''Z'''<sup>''n''</sup> ⋊ SL(''n'', '''Z'''), '''Z'''<sup>''n''</sup>) have relative property (T) for ''n'' ≥ 2.
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| * For ''n'' ≥ 2, the noncompact Lie group Sp(''n'', 1) of isometries of a [[quaternion]]ic [[hermitian form]] of signature (''n'',1) is a simple Lie group of real rank 1 that has property (T). By Kazhdan's theorem, lattices in this group have property (T). This construction is significant because these lattices are [[hyperbolic group]]s; thus, there are groups that are hyperbolic and have property (T). Explicit examples of groups in this category are provided by arithmetic lattices in Sp(''n'', 1) and certain quaternionic [[reflection group]]s.
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| Examples of groups that ''do not'' have property (T) include
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| * The additive groups of integers '''Z''', of real numbers '''R''' and of ''p''-adic numbers '''Q'''<sub>''p''</sub>.
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| * The special linear groups SL(2, '''Z''') and SL(2, '''R'''), although SL(2) has property (T) with respect to principal congruence subgroups, by Selberg's theorem.
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| * Noncompact [[solvable group]]s.
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| * Nontrivial [[free group]]s and [[free abelian group]]s.
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| ==Discrete groups==
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| Historically property (T) was established for discrete groups Γ by embedding them as lattices in real or p-adic Lie groups with property (T). There are now several direct methods available.
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| *The ''algebraic'' method of Shalom applies when Γ = SL(''n'', ''R'') with ''R'' a ring and ''n'' ≥ 3; the method relies on the fact that Γ can be [[boundedly generated group|boundedly generated]], i.e. can be expressed as a finite product of easier subgroups, such as the elementary subgroups consisting of matrices differing from the identity matrix in one given off-diagonal position.
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| *The ''geometric'' method has its origins in ideas of Garland, [[Mikhail Gromov (mathematician)|Gromov]] and [[Pierre Pansu]]. Its simplest combinatorial version is due to Zuk: let Γ be a discrete group generated by a finite subset ''S'', closed under taking inverses and not containing the identity, and define a finite [[Graph (mathematics)|graph]] with vertices ''S'' and an edge between ''g'' and ''h'' whenever ''g''<sup>−1</sup>''h'' lies in ''S''. If this graph is connected and the smallest non-zero eigenvalue of its [[Laplacian matrix|Laplacian]] is greater than ½, then Γ has property (T). A more general geometric version, due to Zuk and {{harvtxt|Ballmann|Swiatkowski|1997}}, states that if a discrete group Γ acts [[properly discontinuous]]ly and [[cocompact]]ly on a [[contractible]] 2-dimensional [[simplicial complex]] with the same graph theoretic conditions placed on the [[link (geometry)|link]] at each vertex, then Γ has property (T). Many new examples of [[hyperbolic group]]s with property (T) can be exhibited using this method.
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| == Applications ==
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| *[[Grigory Margulis]] used the fact that SL(''n'', '''Z''') (for ''n'' ≥ 3) has property (T) to construct explicit families of [[expanding graph]]s, that is, graphs with the property that every subset has a uniformly large "boundary". This connection led to a number of recent studies giving an explicit estimate of ''Kazhdan constants'', quantifying property (T) for a particular group and a generating set.
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| *[[Alain Connes]] used discrete groups with property (T) to find examples of [[von Neumann algebra|type II<sub>1</sub> factors]] with [[countable]] [[von Neumann algebra|fundamental group]], so in particular not the whole of the positive reals. Sorin Popa subsequently used relative property (T) for discrete groups to produce a type II<sub>1</sub> factor with trivial fundamental group.
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| *Groups with property (T) lead to good [[mixing (mathematics)|mixing]] properties in [[ergodic theory]]: again informally, a process which mixes slowly leaves some subsets ''almost invariant''.
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| *Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can efficiently approximate any given invertible matrix, in the sense that every matrix can be approximated, to a high degree of accuracy, by a finite product of matrices in the list or their inverses, so that the number of matrices needed is proportional to the number of [[significant digit]]s in the approximation.
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| *Groups with property (T) also have [[Serre's property FA]].<ref>{{cite journal | first=Yasuo | last=Watatani | title=Property T of Kazhdan implies property FA of Serre. | journal=Math. Japon. | volume=27 | pages=97–103 | year=1981 | zbl=0489.20022 | mr=MR649023 }}</ref>
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| ==References==
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| {{reflist}}
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| *{{citation|first=W.|last= Ballmann |first2=J.|last2= Swiatkowski|doi=10.1007/s000390050022
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| |title=L<sup>2</sup>-cohomology and property (T) for automorphism groups of polyhedral cell complexes|journal= GAFA |volume=7|issue=4|year=1997|pages= 615–645}}
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| *{{Citation | last1=Bekka | first1=Bachir | last2=de la Harpe | first2=Pierre | last3=Valette | first3=Alain | title=Kazhdan's property (T) | publisher=[[Cambridge University Press]] | series=New Mathematical Monographs | isbn=978-0-521-88720-5 | id={{MathSciNet | id = 2415834}} | year=2008 | volume=11|url=http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf }}
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| *{{citation|first= P.|last= de la Harpe |first2= A.|last2= Valette|title=La propriété (T) de Kazhdan pour les groupes localement compactes (with an appendix by M. Burger)|journal= Astérisque |volume=175|year=1989}}.
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| *{{Citation|last=Kazhdan|first=D.|authorlink=David Kazhdan|title=On the connection of the dual space of a group with the structure of its closed subgroups| year = 1967| journal= Functional analysis and its applications| volume = 1|issue=1|pages=63–65|doi=10.1007/BF01075866}}{{MathSciNet|id=0209390}}
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| *{{citation|first=A.|last= [[Alexander Lubotzky|Lubotzky]]|title=Discrete groups, expanding graphs and invariant measures|series=Progress in Mathematics|volume= 125|publisher=Birkhäuser Verlag|year=1994| isbn=3-7643-5075-X|publication-place= Basel }}
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| * [[Alexander Lubotzky|Lubotzky]], A. and A. Zuk, [http://www.ma.huji.ac.il/~alexlub/BOOKS/On%20property/On%20property.pdf ''On property (τ)''], monograph to appear.
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| *{{citation|first=A.|last= [[Alexander Lubotzky|Lubotzky]]|url= http://www.ams.org/notices/200506/what-is.pdf |title=What is property (τ)|journal= AMS Notices|volume= 52 |year=2005|issue= 6|pages=626–627}}.
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| *{{citation|first=Y.|last= Shalom|chapter-url=http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_60.pdf |chapter=The algebraization of property (T)|title= [[International Congress of Mathematicians]] Madrid 2006|year= 2006}}
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| *{{citation|first=A.|last= Zuk|title=La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres|journal=C. R. Acad. Sci. Paris|volume=323|year=1996|pages= 453–458}}.
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| *{{citation|first=A.|last= Zuk|doi=10.1007/s00039-003-0425-8|title=Property (T) and Kazhdan constants for discrete groups |journal=GAFA |volume=13|issue=3|year=2003|pages= 643–670}}.
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| [[Category:Unitary representation theory]]
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| [[Category:Topological groups]]
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| [[Category:Geometric group theory]]
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