On the growth of the wallpaper groups

Abstract

Coordination sequences (also called growth functions) appear in various areas of chemistry and crystallography, such as ice crystals and zeolites, and various areas of mathematics, such as lattice theory and geometric group theory. Cannon's method of cone types is modified for finding the coordination sequence of the Cayley graph of a group. This method is then applied to compute the growth functions and the growth series of the Cayley graphs of the wallpaper groups (the 2D crystallographic groups). The paper has several tables and colored figures summarizing and illustrating the results.