  We prove the vanishing of bounded cohomology with separable dual coefficients
for many groups of interest in geometry, dynamics, and algebra. These include
compactly supported structure-preserving diffeomorphism groups of certain
manifolds; the group of interval exchange transformations of the half line;
piecewise linear and piecewise projective groups of the line, giving strong
answers to questions of Calegari and Navas; direct limit linear groups of
relevance in algebraic K-theory, thereby answering a question by Kastenholz and
Sroka and a question of two of the authors and L\"oh; and certain subgroups of
big mapping class groups, such as the stable braid group and the stable mapping
class group, proving a conjecture of Bowden. Moreover, we prove that in the
recently introduced framework of enumerated groups, the generic group has
vanishing bounded cohomology with separable dual coefficients. At the heart of
our approach is an elementary algebraic criterion called the commuting cyclic
conjugates condition that is readily verifiable for the aforementioned large
classes of groups.
