Boundary groupoids can be used to model many analysis problems on singular spaces. In this talk, we first show that the n-term vanishes for elliptic differential operators on renormalizable boundary groupoids, which is based on the method of renormalized trace similar to that of Moroianu and Nistor. Next we introduce the notion of deformation from the pair groupoid. Under the assumption that a deformation from the pair groupoid M × M exists for Lie groupoid G ⇉ M, we construct explicitly a deformation index map relating the analytic index on G and the index on the pair groupoid. We apply this map to boundary groupoids of the form G = M0 × M0 u G × M1 × M1 ⇉ M = M0 u M1 , where G is an exponential Lie group, to obtain index formulae for (fully) elliptic (pseudo)-differential operators on G , with the aid of the index formula by M. J. Pflaum, H. Posthuma, and X. Tang. It is joint work with Bing Kwan so (Jilin University).