  Let $(G,\mu)$ be a discrete group with a generating probability measure. Nevo
shows that if $G$ has property (T) then there exists an $\epsilon>0$ such that
the Furstenberg entropy of any $(G,\mu)$-stationary ergodic space is either
zero or larger than $\epsilon$.
  Virtually free groups, such as $SL_2(\mathbb{Z})$, do not have property (T),
and neither do their extensions, such as surface groups. For these, we
construct stationary actions with arbitrarily small, positive entropy. This
construction involves building and lifting spaces of lamplighter groups. For
some classical lamplighters, these spaces realize a dense set of entropies
between zero and the Poisson boundary entropy.
