Let G be a group. A Borel
G-space 𝒳 with a σ-finite quasiinvariant measure α is called strongly approximately
transitive (SAT) if there exists an absolutely continuous probability measure ν such
that the closed convex hull co(Gν) of the orbit Gν coincides with the space of
absolutely continuous probability measures on 𝒳. Call a G-space (𝒳,α) purely
atomic if α is purely atomic. Boundaries of stationary random walks on countable G
are always SAT and provide many examples of nonatomic SAT actions. The class of
nonatomic SAT G-spaces also includes certain homogeneous spaces of locally
compact groups. Every countable nonamenable group and also some amenable groups
admit nonatomic SAT actions. However, if G contains a countable nilpotent subgroup
of finite index then every SAT G-space is necessarily purely atomic. This
implies the Choquet-Deny theorem for such groups. Existence of nonatomic
SAT actions is related to growth conditions. A finitely generated solvable
group has polynomial growth if and only if it does not admit nonatomic SAT
actions.
