We consider groups which are the free products with amalgamation of cyclic
groups. These are of the form
/ | p p p \
/ | 1 2 k \
G = / x ,x ,...,x |x = x = ... = x \, where 2 <= p <= p <= ... <= p ;
\ 1 2 k| 1 2 k / 1 2 k
\ | /
\ /
k >= 2 (for infinite cyclic factors), and
/ | rp rp rp p p p \
/ | 1 2 k 1 2 k \
H = / x ,x ,...,x |x = x = ... = x = 1,x = x = ... = x \,
\ 1 2 k| 1 2 k 1 2 k /
\ | /
\ /
where 2 <= p <= p <= ... <= p ; k >= 2; r >= 2 (for finite cyclic factors).
1 2 k
/ | p q\
We begin by looking groups of the form G = / x,y|x = y \, where
\ | /
\ /
2 <= p <= q. This is the free product with amalgamation of two infinite
cyclic groups, and as such there is a canonical (standard) normal form
associated with G. The set of standard normal form words of G is the set
a b a b
mp b 1 1 n n a
of words of the form x y x y ...x y x , where m,n in |Z, -1p < a,a <= 1p,
2 i 2
a != 0, -1q < b,b <= 1q, b != 0. Unless otherwise stated, all quantities
i 2 j 2 j
will be assumed to be integers.
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