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| In [[mathematics]], an '''amenable group''' is a [[locally compact]] [[topological group]] ''G'' carrying a kind of averaging operation on bounded functions that is [[Invariant (mathematics)|invariant]] under translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of ''G'', was introduced by [[John von Neumann]] in 1929 under the [[German language|German]] name "messbar" ("measurable" in English) in response to the [[Banach–Tarski paradox]]. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun.<ref>Day's first published use of the word is in his abstract for an AMS summer meeting in 1949,
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| [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183514222 ''Means on semigroups and groups'', Bull. A.M.S. 55 (1949) 1054–1055]. Many text books on amenabilty, such as Volker Runde's, suggest that Day chose the word as a pun.
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| </ref>
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| The '''amenability''' property has a large number of equivalent formulations. In the field of [[mathematical analysis|analysis]], the definition is in terms of linear functionals. An intuitive way to understand this version is that the [[support (mathematics)|support]] of the [[regular representation]] is the whole space of [[irreducible representation]]s.
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| In [[discrete group theory]], where ''G'' has the [[discrete topology]], a simpler definition is used. In this setting, a group is amenable if one can say what proportion of ''G'' any given subset takes up.
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| If a group has a [[Følner sequence]] then it is automatically amenable.
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| ==Definition for locally compact groups==
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| Let <math>G</math> be a [[locally compact]] [[Hausdorff space|Hausdorff]] [[group (mathematics)|group]]. Then it is well known that it possesses a unique, up-to-scale left- (or right-) rotation invariant ring (borel regular in the case of <math>G</math> second countable) measure (left and right probability measure in the case of <math>G</math> compact), the [[Haar measure]]. Consider the banach space <math>L^{\infty}(G)</math> of essentially-bounded measurable functions within this measure space (which is clearly independent of the scale of the Haar measure).
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| '''Definition 1.'''
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| A linear functional <math>\Lambda\in\operatorname{Hom}(L^{\infty}(G),\mathbf{R})\,</math> is said to be a '''mean''' if <math>\Lambda\,</math> has norm 1 and is non-negative (''i.e.'' <math>f\geq 0\,</math> a.e. implies <math>\Lambda(f)\geq 0\,</math>).
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| '''Definition 2.'''
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| A mean <math>\Lambda\in\operatorname{Hom}(L^{\infty}(G),\mathbf{R})\,</math> is said to be '''left-invariant''' (resp. '''right-invariant''') if <math>\Lambda(g\cdot f)=\Lambda(f)\,</math> all <math>g\in G,f\in L^\infty (G)\,</math> with respect to the left (resp. right) shift action of <math>g\cdot f(x)=f(g^{-1}x)\,</math> (resp. <math>g\cdot f(x)=f(xg^{-1})\,</math>).
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| '''Definition 3.'''
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| A locally compact hausdorff group is called '''amenable''' if it admits a left- (or right-)invariant mean.
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| ==Equivalent conditions for amenability==
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| {{harvtxt|Pier|1984}} contains a comprehensive account of the conditions on a second countable locally compact group ''G'' that are equivalent to amenability:<ref>{{harvnb|Pier|1984}}</ref>
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| *'''Existence of a left (or right) invariant mean on''' ''L''<sup>∞</sup>(''G''). The original definition, which depends on the [[axiom of choice]].
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| *'''Existence of left-invariant states.''' There is a left-invariant state on any separable left-invariant unital C* subalgebra of the bounded continuous functions on ''G''.
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| *'''Fixed-point property.''' Any action of the group by continuous [[affine transformation]]s on a [[convex set|compact convex subset]] of a (separable) [[locally convex topological vector space]] has a fixed point. For locally compact abelian groups, this property is satisfied as a result of the [[Markov–Kakutani fixed-point theorem]].
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| *'''Irreducible dual.''' All irreducible representations are weakly contained in the left regular representation λ on ''L''<sup>2</sup>(''G'').
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| *'''Trivial representation.''' The trivial representation of ''G'' is weakly contained in the left regular representation.
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| *'''Godement condition.''' Every bounded positive-definite measure μ on ''G'' satisfies μ(1) ≥ 0. {{harvtxt|Valette|1998}} improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function ''f'' on ''G'', the function Δ<sup>–½</sup>''f'' has non-negative integral with respect to Haar measure, where Δ denotes the modular function.
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| *'''Day's asymptotic invariance condition.''' There is a sequence of integrable non-negative functions φ<sub>''n''</sub> with integral 1 on ''G'' such that λ(''g'')φ<sub>''n''</sub> − φ<sub>''n''</sub> tends to 0 in the weak topology on ''L''<sup>1</sup>(''G'').
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| *'''Reiter's condition.''' For every finite (or compact) subset ''F'' of ''G'' there is an integrable non-negative function φ with integral 1 such that λ(''g'')φ − φ is arbitrarily small in ''L''<sup>1</sup>(''G'') for ''g'' in ''F''.
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| *'''Dixmier's condition.''' For every finite (or compact) subset ''F'' of ''G'' there is unit vector ''f'' in ''L''<sup>2</sup>(''G'') such that λ(''g'')''f'' − ''f'' is arbitrarily small in ''L''<sup>2</sup>(''G'') for ''g'' in ''F''.
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| *'''Glicksberg−Reiter condition.''' For any ''f'' in ''L''<sup>1</sup>(''G''), the distance between 0 and the closed convex hull in ''L''<sup>1</sup>(''G'') of the left translates λ(''g'')''f'' equals | ∫ f |.
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| *'''[[Følner sequence|Følner condition]].''' For every finite (or compact) subset ''F'' of ''G'' there is a measurable subset ''U'' of ''G'' with finite positive Haar measure such that ''m''(''U'' Δ ''gU'')/m(''U'') is arbitrarily small for ''g'' in ''F''.
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| *'''Leptin's condition.''' For every finite (or compact) subset ''F'' of ''G'' there is a measurable subset ''U'' of ''G'' with finite positive Haar measure such that ''m''(''FU'' Δ ''U'')/m(''U'') is arbitrarily small.
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| *'''Kesten's condition'''. Left convolution on ''L''<sup>2</sup>(''G'') by a probability measure on ''G'' gives an operator of operator norm 1.
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| *'''Johnson's cohomological condition.''' The Banach algebra ''A'' = ''L''<sup>1</sup>(''G'') is [[amenable Banach algebra|amenable as a Banach algebra]], i.e. any bounded derivation of ''A'' into the dual of a Banach ''A''-bimodule is inner.
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| ==Case of discrete groups==
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| The definition of amenability is simpler in the case of a [[discrete group]],<ref>See:
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| *{{harvnb|Greenleaf|1969}}
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| *{{harvnb|Pier|1984}}
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| *{{harvnb|Takesaki|2002a}}
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| *{{harvnb|Takesaki|2002b}}
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| </ref> i.e. a group equipped with the discrete topology.<ref>{{Mathworld|DiscreteGroup|Discrete Group}}</ref>
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| '''Definition.''' A discrete group ''G'' is '''amenable''' if there is a finitely additive [[measure (mathematics)|measure]] (also called a mean) —a function that assigns to each subset of ''G'' a number from 0 to 1—such that
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| # The measure is a '''probability measure''': the measure of the whole group ''G'' is 1.
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| # The measure is '''finitely additive''': given finitely many disjoint subsets of ''G'', the measure of the union of the sets is the sum of the measures.
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| # The measure is '''left-invariant''': given a subset ''A'' and an element ''g'' of ''G'', the measure of ''A'' equals the measure of ''gA''. (''gA'' denotes the set of elements ''ga'' for each element ''a'' in ''A''. That is, each element of ''A'' is translated on the left by ''g''.)
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| This definition can be summarized thus: ''G'' is amenable if it has a finitely-additive left-invariant probability measure. Given a subset ''A'' of ''G'', the measure can be thought of as answering the question: what is the probability that a random element of ''G'' is in ''A''?
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| It is a fact that this definition is equivalent to the definition in terms of ''L''<sup>∞</sup>(''G'').
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| Having a measure <math>\mu</math> on ''G'' allows us to define integration of bounded functions on ''G''. Given a bounded function <math>f:G\to\mathbf{R}</math>, the integral
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| :<math>\int_G f\,d\mu</math>
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| is defined as in [[Lebesgue integration]]. (Note that some of the properties of the Lebesgue integral fail here, since our measure is only finitely additive.)
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| If a group has a left-invariant measure, it automatically has a bi-invariant one. Given a left-invariant measure <math>\mu</math>, the function <math>\mu^-(A)=\mu(A^{-1})</math> is a right-invariant measure. Combining these two gives a bi-invariant measure:
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| :<math>\nu(A) = \int_{g\in G}\mu(Ag^{-1}) \, d\mu^-.</math>
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| The equivalent conditions for amenability also become simpler in the case of a countable discrete group Γ. For such a group the following conditions are equivalent:<ref>{{harvnb|Pier|1984}}</ref>
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| * Γ is amenable.
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| * If Γ acts by isometries on a (separable) Banach space ''E'', leaving a weakly closed convex subset ''C'' of the closed unit ball of ''E''* invariant, then Γ has a fixed point in ''C''.
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| * There is a left invariant norm-continuous functional μ on ''l''<sup>∞</sup>(Γ) with μ(1) = 1 (this requires the [[axiom of choice]]).
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| * There is a left invariant [[C* algebra|state]] μ on any left invariant separable unital [[C* algebra|C* subalgebra]] of ''l''<sup>∞</sup>(Γ).
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| * There is a set of probability measures μ<sub>''n''</sub> on Γ such that ||''g'' · μ<sub>''n''</sub> − μ<sub>''n''</sub>||<sub>1</sub> tends to 0 for each ''g'' in Γ (M.M. Day).
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| * There are unit vectors ''x''<sub>''n''</sub> in ''l''<sup>2</sup>(Γ) such that ||''g'' · ''x''<sub>''n''</sub> − ''x''<sub>''n''</sub>||<sub>2</sub> tends to 0 for each ''g'' in Γ (J. Dixmier).
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| * There are finite subsets ''S''<sub>''n''</sub> of Γ such that | ''g'' · ''S''<sub>''n''</sub> Δ ''S''<sub>''n''</sub> | / |''S''<sub>''n''</sub>| tends to 0 for each ''g'' in Γ (Følner).
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| * If μ is a symmetric probability measure on Γ with support generating Γ, then convolution by μ defines an operator of norm 1 on ''l''<sup>2</sup>(Γ) (Kesten).
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| * If Γ acts by isometries on a (separable) Banach space ''E'' and ''f'' in ''l''<sup>∞</sup>(Γ, ''E''*) is a bounded 1-cocycle, i.e. ''f''(''gh'') = ''f''(''g'') + ''g''·''f''(''h''), then ''f'' is a 1-coboundary, i.e. ''f''(''g'') = ''g''·φ − φ for some φ in ''E''* (B.E. Johnson).
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| * The [[von Neumann algebra|von Neumann group algebra]] of Γ is [[hyperfinite]] (A. Connes).
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| Note that A. Connes also proved that the von Neumann group algebra of any connected locally compact group is [[hyperfinite]], so the last condition no longer applies in the case of locally compact groups.
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| Amenability is related to the spectral problem of Laplacians. For instance, the fundamental group of a closed Riemannian manifold is amenable if and only if the bottom of the spectrum of the Laplacian is 0 (R. Brooks, T. Sunada).
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| ==Properties==
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| *Every (closed) subgroup of an amenable group is amenable.
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| *Every quotient of an amenable group is amenable.
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| *A [[group extension]] of an amenable group by an amenable group is again amenable. In particular, finite [[direct product of groups|direct product]] of amenable groups are amenable, although infinite products need not be.
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| *Direct limits of amenable groups are amenable. In particular, if a group can be written as a directed union of amenable subgroups, then it is amenable.
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| *Amenable groups are [[Uniformly bounded representation|unitarizable]]; the converse is an open problem.
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| *Countable discrete amenable groups obey the [[Ornstein isomorphism theorem]].<ref>D. Ornstein and B. Weiss. "Entropy and isomorphism theorems for actions of amenable groups." ''J. Analyse Math.'' '''48''' (1987), pp.1–141.</ref><ref>Lewis Bowen (2011), "[http://arxiv.org/abs/1103.4424 Every countably infinite group is almost Ornstein]", ArXiv abs/1103.4424</ref>
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| ==Examples==
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| * [[Finite group]]s are amenable. Use the [[counting measure]] with the discrete definition. More generally, [[Compact space|compact]] groups are amenable. The Haar measure is an invariant mean (unique taking total measure 1).
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| * The group of [[integer]]s is amenable (a sequence of intervals of length tending to infinity is a Følner sequence).The existence of a shift-invariant, finitely additive probability measure on the group '''Z''' also follows easily from the [[Hahn–Banach theorem]] this way. Let ''S'' be the shift operator on the [[Sequence_space#.E2.84.93p_spaces|sequence space]] ℓ<sup>∞</sup>('''Z'''), which is defined by (''Sx'')<sub>''i''</sub> = ''x''<sub>''i''+1</sub> for all ''x'' ∈ ℓ<sup>∞</sup>('''Z'''), and let ''u'' ∈ ''ℓ''<sup>∞</sup>('''Z''') be the constant sequence ''u''<sub>''i''</sub> = 1 for all ''i'' ∈ '''Z'''. Any element ''y'' ∈ ''Y'':=Ran(''S'' − ''I'') has a distance larger than or equal to 1 from ''u'' (otherwise ''y<sub>i</sub> = x<sub>i+1</sub> - x<sub>i</sub>'' would be positive and bounded away from zero, whence ''x''<sub>''i''</sub> could not be bounded). This implies that there is a well-defined norm-one linear form on the subspace '''R'''u '''+''' ''Y'' taking ''tu + y'' to ''t''. By the Hahn–Banach theorem the latter admits a norm-one linear extension on ℓ<sup>∞</sup>('''Z'''), which is by construction a shift-invariant finitely additive probability measure on '''Z'''.
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| * By the direct limit property above, a group is amenable if all its [[finitely generated group|finitely generated]] subgroups are. That is, locally amenable groups are amenable.
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| ** By the [[fundamental theorem of finitely generated abelian groups]], it follows that [[abelian group]]s are amenable.
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| * It follows from the extension property above that a group is amenable if it has a finite [[index of a subgroup|index]] amenable subgroup. That is, virtually amenable groups are amenable.
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| * Furthermore, it follows that all [[solvable group]]s are amenable.
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| All examples above are [[Elementary amenable group|elementary amenable]]. The next class of examples below can be used to exhibit non-elementary amenable examples thanks to the existence of groups of [[Grigorchuk group|intermediate growth]].
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| * Finitely generated groups of [[growth rate (group theory)|subexponential growth]] are amenable. A suitable subsequence of balls will provide a Følner sequence.<ref>See:
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| *{{harvnb|Greenleaf|1969}}
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| *{{harvnb|Pier|1984}}
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| *{{harvnb|Takesaki|2002a}}
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| *{{harvnb|Takesaki|2002b}}
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| </ref> | |
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| ==Counterexamples==
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| If a countable discrete group contains a (non-abelian) [[free group|free]] subgroup on two generators, then it is not amenable. The converse to this statement is the so-called [[von Neumann conjecture]], which was disproved by Olshanskii in 1980 using his ''[[Tarski monster]]s''. Adyan subsequently showed that free [[Burnside group]]s are non-amenable: since they are [[periodic group|periodic]], they cannot contain the free group on two generators. These groups are finitely generated, but not finitely presented. However,
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| in 2002 Sapir and Olshanskii found [[finitely presented group|finitely presented]] counterexamples: non-amenable [[finitely presented group]]s that have a periodic normal subgroup with quotient the integers.<ref>{{citation|last=Olshanskii|first= Alexander Yu.|last2= Sapir|first2= Mark V.|
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| title=Non-amenable finitely presented torsion-by-cyclic groups|journal=Publ. Math. Inst. Hautes Études Sci. |volume= 96 |year=2002|pages= 43–169}}</ref>
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| For finitely generated [[linear group]]s, however, the von Neumann conjecture is true by the [[Tits alternative]]:<ref>{{citation|last = Tits|first = J.|title = Free subgroups in linear groups|journal = J. Algebra|volume = 20|year = 1972|pages = 250–270|doi = 10.1016/0021-8693(72)90058-0|issue = 2}}</ref> every subgroup of '''GL'''(''n'',''k'') with ''k'' a field either has a normal solvable subgroup of finite index (and therefore is amenable) or contains the free group on two generators.
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| Although [[Jacques Tits|Tits]]' proof used [[algebraic geometry]], Guivarc'h later found an analytic proof based on [[V. Oseledets]]' [[multiplicative ergodic theorem]].<ref>{{citation|last=Guivarc'h|first=Yves|title= Produits de matrices aléatoires et applications aux propriétés géometriques des sous-groupes du groupes linéaire|
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| journal= Ergod. Th. & Dynam. Sys.|year=1990|volume=10|pages=483–512|doi=10.1017/S0143385700005708|issue=3}}</ref> Analogues of the Tits alternative have been proved for many other classes of groups, such as [[fundamental group]]s of 2-dimensional [[simplicial complex]]es of [[non-positively curved space|non-positive curvature]].<ref>{{citation|first=Werner|last=Ballmann|first2=Michael|last2=Brin|
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| title=Orbihedra of nonpositive curvature|journal=Inst. Hautes Études Sci. Publ. Math.|volume= 82 |year=1995|pages= 169–209|doi=10.1007/BF02698640}}</ref>
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| ==See also==
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| *[[Uniformly bounded representation]]
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| *[[Kazhdan's property (T)]]
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| {{PlanetMath attribution|id=3598|title=Amenable group}}
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| * {{citation|first=F.P.|last= Greenleaf|title=Invariant Means on Topological Groups and Their Applications|publisher= Van Nostrand Reinhold|year=1969}}
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| * {{citation|first=V.|last= Runde|title=Lectures on Amenability|series= Lecture Notes in Mathematics|volume=1774|publisher= Springer|year=2002|isbn=9783540428527}}
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| *{{citation|first= M.|last= Takesaki|title=Theory of Operator Algebras|publisher=Springer|volume=2|year=2002a|isbn=9783540422488}}
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| *{{citation|first= M.|last= Takesaki|title=Theory of Operator Algebras|volume=2 |publisher=Springer|volume=3|year=2002b|isbn=9783540429142}}
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| * {{citation|first=J|last=von Neumann|authorlink=John von Neumann|url= http://matwbn.icm.edu.pl/ksiazki/fm/fm13/fm1316.pdf|title=Zur allgemeinen Theorie des Maßes|journal= [[Fundamenta Mathematicae|Fund. Math.]]|year= 1929|volume=13 |issue= 1|pages= 73−111}}
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| *{{citation|first=Jacques|last=Dixmier|authorlink=Jacques Dixmier|title= C*-algebras (translated from the French by Francis Jellett)|series= North-Holland Mathematical Library|volume=15|publisher=North-Holland|year= 1977}}
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| *{{citation|first=Jean-Paul|last=Pier|title=Amenable locally compact groups|publisher=Wiley|year=1984|zbl=0621.43001|series=Pure and Applied Mathematics}}
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| *{{citation|last=Valette|first= Alain|title=On Godement's characterisation of amenability|journal=Bull. Austral. Math. Soc.|volume= 57 |year=1998|pages= 153–158}}
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| *{{citation|last=Brooks|first= Robert|authorlink=Robert W. Brooks|title=The fundamental group and the spectrum of the laplacian|journal=[[Commentarii Mathematici Helvetici|Comment. Math. Helv.]]|volume= 56 |year=1981|pages= 581–598}}
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| *{{citation|last=Sunada|first= Toshikazu|authorlink=Toshikazu Sunada|title=Unitary representations of fundamental groups and the spectrum of twisted Laplacians|journal=[[Topology (journal)|Topology]]|volume= 28 |year=1989|pages= 125–132}}
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| ==External links==
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| * [http://terrytao.wordpress.com/2009/04/14/some-notes-on-amenability/ Some notes on amenability] by [[Terry Tao]]
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| [[Category:Topological groups]]
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| [[Category:Geometric group theory]]
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