Geodesic
The number of words of a given length in the planar crystallographic groups
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 188-197 
A. V. Shutov. The number of words of a given length in the planar crystallographic groups. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 188-197. http://geodesic.mathdoc.fr/item/ZNSL_2003_302_a12/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

This paper reveals an interrelation between the words in groups and the growth process of the corresponding graphs, based on which exact formulas for the number of words of a given length for all 17 planar crystallographic groups are obtained. Sloane's conjecture on the graphs of these groups is proved.

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