  Recently, Lusztig constructed for each reductive group a partition by unions
of sheets of conjugacy classes, which is indexed by a subset of the set of
conjugacy classes in the associated Weyl group. Sevostyanov subsequently used
certain elements in each of these Weyl group conjugacy classes to construct
strictly transverse slices to the conjugacy classes in these strata,
generalising the classical Steinberg slice, and similar cross sections were
built out of different Weyl group elements by He-Lusztig.
  In this paper we observe that He-Lusztig's and Sevostyanov's Weyl group
elements share a certain geometric property, which we call minimally dominant;
for example, we show that this property characterises involutions of maximal
length. Generalising He-Nie's work on twisted conjugacy classes in finite
Coxeter groups, we explain for various geometrically defined subsets that their
elements are conjugate by simple shifts, cyclic shifts and strong conjugations.
We furthermore derive for a large class of elements that the principal
Deligne-Garside factors of their powers in the braid monoid are maximal in some
sense. This includes those that are used in He-Lusztig's and Sevostyanov's
cross sections, and explains their appearance there; in particular, all
minimally dominant elements in the aforementioned conjugacy classes yield
strictly transverse slices. These elements are conjugate by cyclic shifts,
their Artin-Tits braids are never pseudo-Anosov in the conjectural
Nielsen-Thurston classification and their Bruhat cells should furnish an
alternative construction of Lusztig's inverse to the Kazhdan-Lusztig map and of
his partition of reductive groups.
