1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2.6. Amenable groups 49

Note that this implies the existence of an approximate invariant mean
given by normalized characteristic functions. Indeed, if XF is the charac-
teristic function over F, then 1 } 1 xF E Prob(r) and a computation confirms
that

It turn out that all the definitions above give rise to the same class of
groups. Before the proof, however, a few examples might be nice.
Example 2.6.5 (Elementary amenable groups). It is not hard to see that
finite groups are amenable (take the state which maps X{s} to 1/lrl, for
each group element). So are abelian groups, as the Markov-Kakutani fixed
point theorem easily implies. (There is an alternate proof below.) It is also
true that the class of amenable groups is closed under taking subgroups,
extensions, quotients and inductive limits. (These all make excellent exer-
cises.) Hence anything built out of finite or abelian groups, using the four
operations above, is also amenable; by definition, these are the elementary
amenable groups. In particular, all solvable (hence all nilpotent) groups are
amenable.

Example 2.6.6 (Groups with subexponential growth). A group r is said
to have subexponential growth if limsup IEnll/n = 1 for every finite subset
E c r. (En= {gig2 · · · gn: gi EE}.) It is clear that if a particular finite set
E satisfies the above condition, then every subset F C En will too. Hence
if r is generated by a finite subset E c r as a semigroup, then it suffices to
check the growth condition only for E.

Such groups are amenable. To see this, we construct an increasing se-
quence Eo c E 1 C E2 c · · · of finite subsets of G, whose union equals r,
such that E;;^1 =En, EmEn C Em+n, and liminf IEnJl/n = 1. (Start with
any finite set, keep throwing in group elements, and then take higher pow-
ers as in the definition of subexponential growth.) It turns out that some
subsequence of {En} must be a Folner sequence. Indeed, for any g E Ek, we
have gEn-k c En, and thus JgEn n Enl ;:::: JgEn-kl = IEn-kl· The proof of
the ratio test, from elementary calculus, contains the following general fact:

lim inf _!!:!!.__ :::; lim inf a~! n,

n->oo an-k n->oo

for an ;:::: 0 and any fixed k EN. Applying the reciprocal of this inequality,
we have

. JgEn n Enl. IEn-kl 1. 1
hmsup n->oo I En I ;:::: hmsup n---+oo IE n I 2': imsup n---+oo I En lkfn = 1.

It is a fun combinatorial exercise to show all abelian groups have subexpo-
nential growth.