Renormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids

We consider the index problem of certain boundary groupoids of the form $cG = M _0 times M _0 cup mathbb{R}^q times M _1 times M _1$. Since it has been shown that when $q $ is odd and $geq 3$, $K _0 (C^* (cG)) cong bbZ $, and moreover the $K$-theoretic index coincides with the Fredholm index, in this paper we attempt to derive a numerical formula. Our approach is similar to that of renormalized trace of Moroianu and Nistor cite{Nistor;Hom2}. However, we find that when $q geq 3$, the eta term vanishes, and hence the $K$-theoretic and Fredholm indexes of elliptic (respectively fully elliptic) pseudo-differential operators on these groupoids are given only by the Atiyah-Singer term. As for the $q=1$ case we find that the result depends on how the singularity set $M_1$ lies in $M$.