  This paper is a merge of arXiv:1807.05418 and arXiv:1808.01442.
  We introduce a new class of groupoids, called "boundary action groupoids",
which are obtained by gluing reductions of action groupoids. We show that such
groupoids model the analysis on many singular spaces, and we give several
examples. Under some conditions on the action of the groupoid, we obtain
Fredholm criteria for the pseudodifferential operators generated by boundary
action groupoids. Moreover, we show that layer potential groupoids for conical
domains constructed in an earlier paper (Carvalho-Qiao, Central European J.
Math., 2013) are both Fredholm groupoids and boundary action groupoids, which
enables us to deal with many analysis problems on singular spaces in a unified
way. As an application, we obtain Fredholm criteria for operators on layer
potential groupoids.
