  The n-strand braid group can be defined as the fundamental group of the
configuration space of n unlabeled points in a closed disk based at a
configuration where all n points lie in the boundary of the disk. Using this
definition, the subset of braids that have a representative where a specified
subset of these points remain pointwise fixed forms a subgroup isomorphic to a
braid group with fewer strands. In this article, we generalize this phenomenon
by introducing the notion of boundary braids. A boundary braid is a braid that
has a representative where some specified subset of the points remains in the
boundary cycle of the disk. Although boundary braids merely form a subgroupoid
rather than a subgroup, they play an interesting geometric role in the
piecewise Euclidean dual braid complex defined by Tom Brady and the second
author. We prove several theorems in this setting, including the fact that the
subcomplex of the dual braid complex determined by a specified set of boundary
braids metrically splits as the direct metric product of a Euclidean polyhedron
and a dual braid complex of smaller rank.
