  In 1965, Steinberg's study of conjugacy classes in connected reductive groups
led him to construct an affine subspace parametrising regular conjugacy
classes, which he noticed is also a cross section for the conjugation action by
the unipotent radical of a Borel subgroup on another affine subspace. Recently,
generalisations of this slice and its cross section property have been obtained
by Sevostyanov in the context of quantum group analogues of W-algebras and by
He-Lusztig in the context of Deligne-Lusztig varieties. Such slices are often
thought of as group analogues of Slodowy slices.
  In this paper we explain their relationship via common generalisations
associated to Weyl group elements and provide a simple criterion for cross
sections in terms of roots. In the most important class of examples this
criterion is equivalent to a statement about the Deligne-Garside factors of
their powers in the braid monoid being maximal in some sense. Moreover, we show
that these subvarieties transversely intersect conjugacy classes and determine
for a large class of factorisable r-matrices when the Semenov-Tian-Shansky
bracket reduces to a Poisson structure on these slices.
