172 5. Exact Groups and Related TopicsFix NE N and define m: lF2-+ Prob(lF2) by
l N-1
mt= NL Ot(k)·
k=O
Clearly, all mt are supported on the finite set of elements in lF 2 with length <
N. An instructive calculation left to the reader confirms that lls.mt-mstll :::;
2lsl/N for every s, t E lF2. Letting N-+ oo completes the proof. 0
The proof above is more transparent when viewed geometrically in the
Cayley graph of JF 2. Moreover, a geometric point of view will be crucial
in the next two sections, so let's develop our intuition by proving that free
groups act amenably on their ideal boundaries.
Fix r E N and let lFr = (gi, ... , 9r) be the rank-r free group (think of
the r = 2 case for now). Then, its ideal boundary is the set
81Fr = {(ai) E IJ{g1,g11, ... ,gr,g;^1 }: Vi E N,ai+l i= ai^1 }.
N.The complement of 81Fr (in IJN{gl, g1\ ... , 9r, g;^1 }) is easily seen to be
open in the product topology; hence 81Fr is compact. For geometric intuition,
it is better to identify 8JF r with the set of infinite paths in the Cayley graph
of lFr which start at the neutral element. Indeed, given x = (xi) E 81Fr, we
first think of x as the infinite word x1x 2 x 3 · · · (note that this is in reduced
form, since no cancellation occurs); then we identify this word with the path
determined by the sequence of vertices {x 1 , x1x 2 , x1x 2 x 3 , ... } in the Cayley
graph of IB'r. Thinking of 81Fr as infinite reduced words, it is easy to see that
lF r acts continuously on 8JF r by left multiplication (and rectifying possible
cancellation).
For x E 8lB'r with reduced word form x = x 1 x 2 ···,we set x(O) = e and
x(k) = x1 · · ·xk for all k > 0. Fix N E N and define a continuous map
μ: 8lF r -+ Prob(lF r) by
N-1
μx = ~ L Ox(k)·
k=O
Looking at the Cayley graph in Figure 1, μx is just the normalized charac-
teristic function of the first N steps along the infinite path determined by
x. Here's an important observation/exercise: For each s E lFr and x E 81Fr,
there exists a unique geodesic path starting at s and eventually merging
with the path determined by s.x E 81Fr (see Figure 2); moreover, s.μx is
just the normalized characteristic function of the first N steps along this
geodesic.