We present a wavelet-based geometry compression pipeline in the context of hierarchical surface and volume representations. Due to the increasing complexity of geometric models used in a vast number of different application field, new methods have to be devised that enable one to store, transmit and manipulate large amounts of data. Based on a multiresolution wavelet representation, we have developed a complete compression pipeline suitable for geometric data on uniform grids in two and three dimensions. Local and global oracles in wavelet space are employed to control the approximation error in lossy compression settings. Two geometry simplification schemes, which are able to build hierarchical mesh representations, are an essential part of the pipeline. In contrast to the two wavelet-based approximation schemes with each other using several two- and three-dimensional data sets and provide an extensive error and performance analysis. These results emphasize the individual strengths and weaknesses of each of the discussed methods and concepts.