- Kazhdan's property (T)
In
mathematics , alocally compact topological group "G" has property (T) if thetrivial representation is anisolated point in itsunitary dual equipped with the Fell topology. Informally, this means that if "G" acts unitarily on aHilbert space and has "almost invariant vectors", then it has a nonzeroinvariant vector . The formal definition, introduced byDavid Kazhdan (1967), gives this a precise, quantitative meaning.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 togroup representation theory , lattices in algebraic groups over local fields,ergodic theory ,geometric group theory ,operator algebras and the theory of networks.Definitions
Let "G" be a locally compact
topological group and π : "G"ightarrow "U"("H") aunitary representation of "G" on a 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 π("g") ξ - ξ < ε for all "g" in "K".The following conditions on "G" are all equivalent to "G" having property (T) of Kazhdan, and any of them can be used as the definition of property (T).
* The
trivial representation is anisolated point of theunitary dual of "G" withFell topology .* Any sequence of continuous positive definite functions on "G" converging to 1 uniformly on
compact subsets, converges to 1 uniformly on "G".* 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.* 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.
* Every continuous affine isometric action of "G" on a real
Hilbert space has a fixed point (property (FH)).If "H" is a
closed subgroup of "G", the pair ("G","H") is said to have relative property (T) of 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".General properties
* Property (T) is preserved under quotients: if "G" has property (T) and "H" is a
quotient group of "H" 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).
* If "G" has property (T) then "G"/[ "G", "G"] is compact.
* Any countable discrete group with property (T) is finitely generated.
* Anamenable group which has property (T) is necessarilycompact . Amenability and property (T) are in a rough sense opposite: they make almost invariant vectors easy or hard to find.
* Kazhdan's theorem: If "Γ" is a 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).Examples
*
Compact topological group s have property (T). In particular, thecircle group , the additive group Z"p" of "p"-adic integers, compactspecial unitary group s "SU"("n") and all finite groups have property (T).
* Simple realLie group s of real rank at least two have property (T). This family of groups includes thespecial linear group s "SL""n"(R) for "n" ≥ 3 and the specialorthogonal group s "SO"("p","q") for "p" > "q" ≥ 2 and "SO"("p","p") for "p" ≥ 3. More generally, this holds for simplealgebraic group s of rank at least two over alocal field .
* The pairs (R"n" times "SL""n"(R), R"n") and (Z"n"times "SL""n"(Z), Z"n") have relative property (T) for "n" ≥ 2.
* For "n" ≥ 2, the noncompact Lie group "Sp"("n",1) of isometries of aquaternion ichermitian 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 arehyperbolic 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 quaternionicreflection group s.Examples of groups that "do not" have property (T) include
* The additive groups of integers Z, of real numbers R and of "p"-adic numbers Q"p".
* The special linear groups "SL"2(Z) and "SL"2(R).
* Noncompactsolvable group s.
* Nontrivialfree group s andfree abelian group s.Discrete groups
Historically property (T) was established for discrete groups Gamma by embedding them as lattices in real or p-adic Lie groups with property (T). There are now several direct methods available.
*The "algebraic" method of Shalom applies when Gamma= SLn("R") with "R" a ring and "n"≥ 3; the method relies on the fact that Gamma can be 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.
*The "geometric" method has its origins in ideas of Garland, Gromov, and
Pierre Pansu . Its simplest combinatorial version is due to Zuk: let Gamma be a discrete group generated by a finite subset "S", closed under taking inverses and not containing the identity, and define a finitegraph with vertices "S" and an edge between "g" and "h" whenever "g"−1 "h" lies in "S". If this graph is connected and the smallest non-zero eigenvalue of its Laplacian is greater than ½, then Gamma has property (T). A more general geometric version, due to Zuk and harvtxt|Ballmann|Swiatkowski|1997, states that if a discrete group Gamma actsproperly discontinuous ly andcocompact ly on acontractible 2-dimensionalsimplicial complex with the same graph theoretic conditions placed on the link at each vertex, then Gamma has property (T). Many new examples ofhyperbolic group s with property (T) can be exhibited using this method.Applications
*
Grigory Margulis used the fact that SL"n"(Z) (for "n"≥3) has property (T) to construct explicit families ofexpanding 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.*
Alain Connes used discrete groups with property (T) to find examples of type II1 factors withcountable 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 II1 factor with trivial fundamental group.*Groups with property (T) lead to good mixing properties in
ergodic theory : again informally, a process which mixes slowly leaves some subsets "almost invariant".*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.References
*citation|first=W.|last= Ballmann |first2=J.|last2= Swiatkowski|doi=10.1007/s000390050022
title=L2-cohomology and property (T) for automorphism groups of polyhedral cell complexes|journal= GAFA |volume=7|year=1997|pages= 615-645
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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.
*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|pages=63-65|doi=10.1007/BF01075866MathSciNet|id=0209390
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* Lubotzky, A. and A. Zuk, [http://www.ma.huji.ac.il/~alexlub/BOOKS/On%20property/On%20property.pdf "On property (τ)"] , monograph to appear.
*citation|first=A.|last= 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.
*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
*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.
*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|year=2003|pages= 643-670.
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