Certain HNN-extensions of cyclic groups.

Consider the group


                  /   | -1 p     p  rp    \
             G = / x,y|y  x y = x ,x   = 1 \    (p >= 2,r >= 2).
                 \    |                    /
                  \                       /

Then G is an HNN-extension of the cyclic  group  of  order  rp  and  so  is
automatic.

We shall show that the  growth  series  of  G  with  respect  to  {x,y}  is
rational.  It turns out however that if r = 2 and we let t = 1, or if p = 2
and we let t = 1 then  g(1) != __1_.
                               x(G)

                                                             (       p)
We then show that the growth series g' of G with respect to  |x,y,z=x |  is
                                                             (        )

rational  and  for this generating set we get the identity g'(1) = __1_ for
                                                                   x(G)

r >= 2.  If r = 2 and p is odd our calculation of  g' agrees  with  that  of
I.M.Chiswell.

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