  Boundary groupoids were introduced by the second author, which can be used to
model many analysis problems on singular spaces. In order to investigate index
theory on boundary groupoids, we introduce the notion of {\em a deformation
from the pair groupoid}.Under the assumption that a deformation from the pair
groupoid $M \times M$ exists for Lie groupoid $\mathcal{G}\rightrightarrows M$,
we construct explicitly a deformation index map relating the analytic index on
$\mathcal{G}$ and the index on the pair groupoid. We apply this map to boundary
groupoids of the form $\mathcal{G} = M_0 \times M_0 \sqcup G \times M_1 \times
M_1 \rightrightarrows M=M_0\sqcup M_1$, where $G$ is an exponential Lie group,
to obtain index formulae for (fully) elliptic (pseudo)-differential operators
on $\mathcal{G}$, with the aid of the index formula by M. J. Pflaum, H.
Posthuma, and X. Tang. These results recover and generalize our previous
results for renormalizable boundary groupoids via the method of renormalized
trace.
