The Euler class is
a semiconjugacy invariant of a discrete group G of orientation preserving
homeomorphisms of the circle. An element of the second cohomology group of G with
integral coefficients, it is often difficult to calculate, but even its nonvanishing seems
related to dynamical complexity of G. In this note, we consider a family of
discrete groups ΓH,S(p,q) of homeomorphisms of the circle, whose definition
generalizes that of piecewise linear homeomorphisms. We define an invariant
with which one can verify the vanishing of the Euler class in a surprising
range of cases. On the other hand, the vanishing of the invariant, together
with a simple geometric condition, assures the nonvanishing of the Euler
class.
