In this paper we find all solvable subgroups of
and classify their actions.
We also investigate the
local rigidity of actions of the solvable Baumslag–Solitar groups on the circle.
The investigation leads to two novel phenomena in the study of infinite
group actions on compact manifolds. We exhibit a finitely generated group
has exactly countably infinitely many effective real-analytic actions on
, up to
(ii) every effective, real analytic action of
for every such
there are infinitely many nonconjugate, effective real-analytic actions of