Densities in free groups and Z k , Visible Points and Test Elements
Abstract
In this article we relate two different densities. Let
F
k
be the free group of finite rank
k≥2
and let
α
be the abelianization map from
F
k
onto
Z
k
. We prove that if
S⊆
Z
k
is invariant under the natural action of
SL(k,Z)
then the asymptotic density of
S
in
Z
k
and the annular density of its full preimage
α
−1
(S)
in
F
k
are equal. This implies, in particular, that for every integer
t≥1
, the annular density of the set of elements in
F
k
that map to
t
-th powers of primitive elements in
Z
k
is equal to to
1
t
k
ζ(k)
, where
ζ
is the Riemann zeta-function. An element
g
of a group
G
is called a \emph{test element} if every endomorphism of
G
which fixes
g
is an automorphism of
G
. As an application of the result above we prove that the annular density of the set of all test elements in the free group
F(a,b)
of rank two is
1−
6
π
2
. Equivalently, this shows that the union of all proper retracts in
F(a,b)
has annular density
6
π
2
. Thus being a test element in
F(a,b)
is an ``intermediate property'' in the sense that the probability of being a test element is strictly between 0 and 1.