# Amenable group

In

mathematics , an**amenable group**is alocally compact topological group "G" carrying a kind of averaging operation on bounded functions that isinvariant under left (or right) translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of "G", was introduced byJohn von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to theBanach-Tarski paradox . In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun. [*Day's first published use of the word is in his abstract for an AMS summer meeting in 1949, [*]*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.The

**amenability**property has a large number of equivalent formulations. In the field of analysis, the definition is in terms of linear functionals. An intuitive way to understand this version (which can be made precise) is that the support of theregular representation is the whole space ofirreducible representation s.In

discrete group theory , where $G$ has no topological structure, a simpler definition is used. In this setting, a group is amenable if one can say what percentage of $G$ any given subset takes up.If a group has a

Følner sequence then it is automatically amenable.**Locally compact definition**Let $G$ be a

locally compact group and $L^infty(G)$ be theBanach space of all essentially bounded functions $G\; o\; Bbb\{R\}$ with respect to theHaar measure .**Definition 1.**A linear functional on $L^infty(G)$ is called a "mean" if it maps the constant function $f(g)\; =\; 1$ to 1 and non-negative functions to non-negative numbers.**Definition 2.**Let $L\_g$ be the left action of $g\; in\; G$ on $f\; in\; L^infty(G)$,i.e. $(L\_g\; f)(h)\; =\; f(g^\{-1\}h)$.Then, a mean $mu$ is said to be "left-invariant" if $mu(L\_g\; f)\; =\; mu(f)$for all $g\; in\; G$ and $f\; in\; L^infty(G).$Similarly, $mu$ is said to be "right-invariant" if $mu(R\_g\; f)\; =\; mu(f),$where $R\_g$ is the right action $(R\_g\; f)(h)\; =\; f(hg).$**Definition 3.**A locally compact group $G$ is**amenable**if there is a left- (or right-)invariant mean on $L^infty(G).$**Discrete definition**The definition of amenability is quite a lot simpler in the case of a

discrete group , i.e. a group with no topological structure.**Definition.**A discrete group $G$ is**amenable**if there is a measure—a function that assigns to each subset of $G$ a number from 0 to 1—such that# The measure is a

**probability measure**: the measure of the whole group $G$ is 1.

# 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.

# 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 $g$.)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 $A$?

It is a fact that this definition is equivalent to the definition in terms of $L^infty(G)$.

Having a measure $mu$ on $G$ allows us to define integration of bounded functions on $G$. Given a bounded function $f:G\; omathbf\{R\}$, the integral:$int\_G\; f,dmu$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.)If a group has a left-invariant measure, it automatically has a bi-invariant one. Given a left-invariant measure $mu$, the function $mu^-(A)=mu(A^\{-1\})$ is a right-invariant measure. Combining these two gives a bi-invariant measure::$u(A)=int\_\{gin\; G\}mu(Ag^\{-1\})dmu^-.$

**Conditions for a discrete group**The following conditions are equivalent for a countable discrete group Γ:

* Γ is amenable.

* 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".

* There is a left invariant norm-continuous functional μ on "l"^{∞}(Γ) with μ(1) = 1 (this requires theaxiom of choice ).

* There is a left invariant state μ on any left invariant separable unital C* subalgebra of "l"^{∞}(Γ).

* There is a set of probability measures μ_{"n"}on Γ such that ||"g" · μ_{"n"}- μ_{"n"}||_{1}tends to 0 for each "g" in Γ (M.M. Day).

* There are unit vectors "x"_{"n"}in "l"^{2}(Γ) such that ||"g" · "x"_{"n"}- "x"_{"n"}||_{2}tends to 0 for each "g" in Γ (J. Dixmier).

* There are subsets "S"_{"n"}of Γ such that | "g" · "S"_{"n"}Δ "S"_{"n"}| / |"S"_{"n"}| tends to 0 for each "g" in Γ (Følner).

* 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"^{2}(Γ) (Kesten).

* If Γ acts by isometries on a (separable) Banach space "E" and "f" in "l"^{∞}(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).

* The von Neumann group algebra of Γ ishyperfinite (A. Connes).**Examples***

Finite group s are amenable. Use thecounting measure with the discrete definition.

*Subgroup s of amenable groups are amenable.

* Thedirect product of two amenable groups is amenable, while thedirect product of an infinite family of amenable groups need not be.

* The group ofinteger s is amenable (they have aFølner sequence ).

* A group is amenable if all its finitely generated subgroups are. That is, locally amenable groups are amenable.

** By thefundamental theorem of finitely generated abelian groups , it follows thatabelian group s are amenable.

* A group is amenable if it has an amenablenormal subgroup such that the quotient is amenable. That is, extensions of amenable groups by amenable groups are amenable.

** It follows that a group is amenable if it has a finite index amenable subgroup. That is, virtually amenable groups are amenable.

** Furthermore, it follows that allsolvable group s are amenable.

* Compact groups are amenable. The Haar measure is an invariant mean (unique taking total measure 1).

* Finitely generated groups of subexponential growth are amenable.**Non-examples**If a countable discrete group contains a (non-abelian) 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 monsters". Adyan subsequently showed that freeBurnside group s are non-amenable: since they are periodic, they cannot contain the free group on two generators. In 2002, Sapir and Olshankii found finitely generated counterexamples: non-amenablefinitely presented group s that have periodic normal subgroups of finite index. [*citation|last=Olshanskii|first= Alexander Yu.|last2= Sapir|first2= Mark V.*]

title=Non-amenable finitely presented torsion-by-cyclic groups|journal=Publ. Math. Inst. Hautes Études Sci. |volume= 96 |year=2002|pages= 43-169For finitely generated

linear group s, however, the von Neumann conjecture is true by theTits alternative [*citation|last = Tits|first = J.|title = Free subgroups in linear groups|journal = J. Algebra|volume = 20|date = 1972|pages = 250-270*] : 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.Although Tits' proof usedalgebraic geometry , Guivarc'h later found an analytic proof based on Oseledets'multiplicative ergodic theorem . [*citation|last=Guivarc'h|first=Yves|title= Produits de matrices aléatoires et applications aux propriétés géometriques des sous-groupes du groupes linéaire*] Analogues of the Tits alternative have been proved for many other classes of groups, such as

journal= Ergod. Th. & Dynam. Sys.|year=1990|volume=10|pages=483-512fundamental group s of 2-dimensionalsimplicial complex es of non-positive curvature. [*citation|first=Werner|last=Ballmann|first2=Michael|last2=Brin| title=Orbihedra of nonpositive curvature|journal=Inst. Hautes Études Sci. Publ. Math.|volume= 82 |year=1995|pages= 169-209*]**ee also***

amenable Banach algebra **Notes****References*** F.P. Greenleaf, "Invariant Means on Topological Groups and Their Applications", Van Nostrand Reinhold (1969).

* V. Runde, "Lectures on Amenability", Lecture Notes in Mathematics**1774**, Springer (2002).

* M. Takesaki, "Theory of Operator Algebras", Vol. 2 and 3, Springer.

* J. von Neumann, "Zur allgemeinen Theorie des Maßes", Fund. Math.**13**(1929), 73−111. JFM|55.0151.01----

*planetmath|id=3598|title=Amenable group

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