We introduce a new variant of the popular Burrows-Wheeler transform (BWT) called Geometric Burrows-Wheeler Transform (GBWT). Unlike BWT, which merely permutes the text, GBWT converts the text into a set of points in 2-dimensional geometry. Using this transform, we cananswer to many open questions in compressed text indexing: (1) Can compressed data structures be designed in external memory with similar performance as the uncompressed counterparts? (2) Can compressed data structures be designed for position restricted pattern matching? We also introduce a reverse transform, called??Points2Text, which converts a set of pointsinto text. This transform allows us to derive the FIRST known lower bounds in compressed text indexing. We show strong equivalence between data structural problems in geometric range searching and text pattern matching. This provides a way to derive new results in compressed text indexing by translating the results from range searching.