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We introduce the concept of Roe C*-algebra for a locally compact groupoidwhose unit space is in general not compact, and that is equipped with anappropriate coarse structure and Haar system. Using Connes' tangent groupoidmethod, we introduce an analytic index for an elliptic differential operator ona Lie groupoid equipped with additional metric structure, which takes values inthe K-theory of the Roe C*-algebra. We apply our theory to derive aLichnerowicz type vanishing result for foliations on open manifolds.

Roe C*-algebra for groupoids and generalized Lichnerowicz Vanishing theorem for foliated manifolds