We introduce two transformation Text2Points and Points2Text which converts text to points in space and vice-versa. With these transformations, the data structural problems in pattern matching and range searching can be linked. We show strong connections between space vs query-time trade-offs in these fields. Thus, results in range searching can be applied to compressed text indexing and vice-versa. In particular, we show that for a given equivalent space, pattern matching queries can be done using 2-d range searching and vice-versa with query times within factor of $O(\log n)$ of each other. This two-way connection, enables us not only to design new data structures for compressed text indexing but also to derive new lower bounds. For compressed text indexing, we propose the alternative data structures based on our Text2Points transform and 4-sided orthogonal query structures in 2-d. Currently, all proposed compressed text indexes are based on Burrows Wheeler Transform (BWT) \cite{gv,fm,ggv,sada,mn}. We observe that our Text2Points transfer is equivalent to BWT on blocked text, and hence we call it Geometric BWT. With this variant, we can solve some well-known open problems in this area of compressed text indexing. In particular, we present the first external memory results for compressed text indexing. We also give first compressed data strcutures for the position restricted pattern matching\cite{mn-latin,tech}. We also show lower bounds for these problems and for the problem of text indexing in general using our transform Points2Text. These are the first known lower bounds (hardness results) in this area.