  We define a class of transversal slices in spaces which are quasi-Poisson for
the action of a complex semisimple group G. This is a multiplicative analogue
of Whittaker reduction. One example is the multiplicative universal centralizer
of G, which is equipped with the usual symplectic structure in this way. We
construct a smooth relative compactification by taking the closure of each
centralizer fiber in the wonderful compactification of G. By realizing this
partial compactification as a transversal in a larger quasi-Poisson variety, we
show that it is smooth and log-symplectic.
