Concerning certain Baumslag-Solitar groups.

Let


                           /     -1      2k\
                     G  = / x,y|x  yx = y   \  (k >= 1)
                      e   \                 /
                           \               /

and


                          /     -1      2k+1\
                    G  = / x,y|x  yx = y     \  (k >= 1)
                     o   \                   /
                          \                 /


Theorem

(i) If k >= 2 then the growth series of G  with respect to {x,y} is
                                         e

               3        2   3  4   k+1   k+2   k+3   k+4   k+5   2k+2
g  = (1+t)(1-t) (1+2t+2t +2t +t +2t   -2t   +2t   +2t   +4t   -6t
 e

         2k+5   2k+8   3k+4   3k+6   3k+8   4k+5   4k+6   4k+7   4k+8   5k+7 /
     -12t    +2t    -8t    -4t    -4t    +8t    +4t    -8t    +4t    +8t    /
                                                                           /
            2  3   k+2   k+4   2k+4 2       2   k+1   k+2   2k+2
     ((1-t-t -t +2t   -2t   +2t    ) (1-2t-t +2t   -2t   +2t    ))


(ii) If k = 1 then the growth series of G  with respect to {x,y} is
                                         e

             2     2        2    3    4    5    6    7    8    9   10
   g  = (1-t)_(1+t)_(1+3t+8t_+12t_+16t_+20t_+22t_+16t_+14t_+12t_+4t__).
    e                               3     2   5 2
                             (1-t-2t )(1-t -2t )


(iii) The growth series of G  with respect to {x,y} is
                            o

                      (   2   k+2)(      k+2)     2     3
                      |1+t -2t   ||1+t-2t   |(1+t) (1-t)
                 g  = (__________)(_________)____________
                  o                     2
                       (     2  3   k+3) (      2   k+2)
                       |1-t-t -t +2t   | |1-2t-t +2t   |
                       (               ) (             )

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