Deformation and $K$-theoretic Index Formulae on Boundary Groupoids

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.