  For an arbitrary countable discrete infinite group $G$, nonsingular rank-one
actions are introduced. It is shown that the class of nonsingular rank-one
actions coincides with the class of nonsingular $(C,F)$-actions. Given a
decreasing sequence $\Gamma_1\supsetneq\Gamma_2\supsetneq\cdots$ of cofinite
subgroups in $G$ with $\bigcap_{n=1}^\infty\bigcap_{g\in
G}g\Gamma_ng^{-1}=\{1_G\}$, the projective limit of the homogeneous $G$-spaces
$G/\Gamma_n$ as $n\to\infty$ is a $G$-space. Endowing this $G$-space with an
ergodic nonsingular nonatomic measure we obtain a dynamical system which is
called a nonsingular odometer. Necessary and sufficient conditions are found
for a rank-one nonsingular $G$-action to have a finite factor and a nonsingular
odometer factor in terms of the underlying $(C,F)$-parameters. Similar
conditions are also found for a rank-one nonsingular $G$-action to be
isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact
Cantor models are constructed for the nonsingular rank-one extensions of
odometers. Several concrete examples are constructed and several facts are
proved that illustrate a sharp difference of the nonsingular noncommutative
case from the classical finite measure preserving one: odometer actions which
are not of rank one, factors of rank-one systems which are not of rank-one,
however each probability preserving odometer is a factor of an infinite measure
preserving rank-one system, etc.
