1. Show that if an action of a discrete group G on a compact space X is minimal and
locally contracting, then there is no G-invariant probability measure on X. In particular, G
is nonamenable.
2. Consider the action of F2 on its boundary ∂F2 .
(i) Show that the action is minimal and locally contracting.
(ii) Show that every element g ∈ F2 , g 6= e, has exactly two fixed points in ∂F2 . Hint:
if g is such that `(hgh−1 ) ≥ `(g) for all h then these two points are g + = gg . . . and
g − = g −1 g −1 . . . .
(iii) Show that the action is amenable by considering the functions ξnx ∈ `2 (F2 ), x ∈ ∂F2 ,
defined by
(
n−1/2 , if g = x1 . . . xi for some 1 ≤ i ≤ n,
x
ξn (g) =
0,
otherwise.
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