Appendix to the paper "Randomly Growing Braid on Three Strands and the   Manta Ray"
  This paper is an appendix to the paper ``Randomly Growing Braid on Three Strands and the Manta Ray'' by J. Mairesse and F. Math\'eus (to appear in the Annals of Applied Probability). It contains the details of some computations, and the proofs of some results concerning the examples treated there, as well as some extensions. 
Jean Mairesse and Fr\'ed\'eric Math\'eus