5.3.1 Definition A q u o tien t sy ste m A = ( ( X, 7 Z) , ( A, S , w ) , cr) is called a weighted nilpotent quotient s y stem if (A ,c>,iu) is a w eig hted p oly cy clic p re se n ta tio n such th a t th e following co n d itio n s hold.
- For each g e n e ra to r a £ A th e re is e ith e r
- an elem en t x £ X such th a t xcr — a is th e d efin itio n of a in w hich case
w( a) = 1 or
- a p a ir x , y 6 A w ith w ( x ) = 1 such th a t x ~ l y x = y a is th e d efin itio n of
a in w hich case w( a) = u>(y) + 1.
- If A is co n siste n t, th e w eight fu n c tio n w coincides on A w ith th e w eight fu n c tio n defined by th e lower c e n tra l series of th e g ro u p defined by (A,<S).
W e will now d escrib e an a lg o rith m th a t c o n s tru c ts a co n siste n t w eigh ted nilpo- te n t q u o tie n t sy ste m for G / ^ c { G ) .
W e s ta r t w ith Ao = ( (X , 7£), (0, 0), cr0), th e triv ia l q u o tie n t sy ste m b ased on
( X, TZ) . L et X be th e set { z ! , . . ., x m }. T h e n th e c e n tra l covering s y ste m A!Q of
Ao h as th e fo rm ( ( X , 7Z), (A'0, Sq, iv'0), cr'0) w here
A
q -- {&1 , . . . , d m }S'0 = { a ~ l ajCLi = dj I 1 < i < j < m }
wq : a t I— > 1 for 1 < i < m
ctq : Xi I----* di for 1 < i < m .
T h e polycyclic p re s e n ta tio n is co n sisten t by T h e o re m 3.2.3. In o rd e r to m ake A'0
co n siste n t th e re la tio n s in 7Z have to be en forced. By T h e o re m 4.4 th is gives a g e n e ra tin g set { r ( a 1?. . . , a n ) \ r £ TZ } for a s u b g ro u p V of th e free a b e lia n g ro u p
U defined by ( Aq, Sq, w'q ). Let I'Q C { 1 , . . . , n } a n d { a - 1' = w it \ i 6 I'Q } be a set of re la tio n s for U j V w ith a re la tio n m a trix in row H erm ite n o rm a l form .
Now let
I\
CI'Q
be th e indices of g e n e ra to rs for w hichmi
> 1 a n d defineJ
=
{ 1, . .. ,n }\I'Q
UI\.
T h en we get th e follow ing co n siste n t w eig h ted n ilp o te n t q u o tie n t sy ste m A \ = ((Ar , 71), ( A i , S i, W\ ), o q ) re p re se n tin g th e la rg e st ab elian q u o tie n t of G w ith A \ —■■ { dr 1 i 6 J }, S \ =- { a 71 CljCli = <2 j | i, j G J , i < J } U { a™* - w u i ^ }>V>1
: di H-—> 1 for i e J , : Xi i-—> a t for 2 e J , —> w a for i e i ' o V i.L et
Ac =
( ( X , 7 Z ) , ( A c, S c, u>c), crc) be a co n siste n t w eig h ted n ilp o te n t q u o tien t sy ste m re p re se n tin gG / j c-hi(G).
We form th e c e n tra l covering sy ste mA'c
defined by ((X , 71), (A'c, S'c), a'c) as in C h a p te r 4 re p re se n tin g G /j c-\-2{G). L et T =be th e set of new ly defined g en era to rs. T h e w eight fu n c tio n w c is e x te n d e d to a w eight fu n c tio n w'c on A'c by defining w'c( a) = w c( a) if a £ A c a n d w'c(a) = c + 1
if a £ T. T h e n (A ’c, S'c, w'c) is a w eighted n ilp o te n t p re s e n ta tio n . W e define th e follow ing su b se ts of T. Let
- Tc-(-i C T be th e set of new ly defined g e n e ra to rs th a t are defined by a co n ju g a te re la tio n w ith left h a n d side a T 1 ajcq such th a t w{a{) — 1 a n d w ( a j) = c; - Tpc C T be th e set of g e n e ra to rs defined by a pow er re la tio n or a co n ju g ate
re la tio n w ith left h a n d side aj di such th a t w [ a{) = 1; - T a C T be th e set of g e n e ra to rs defined by an im ag e of cr'c \
~~ Tconj Q T be th e set of g e n e ra to rs th a t are defined by a c o n ju g a te re la tio n b u t are n o t elem ents of Tpc.
N ote th a t Tc+1 C Tpc. We define th e lin e a r o rd e r on th e g e n e ra to rs in T such th a t all g e n e ra to rs in Tc+1 are la rg e r th a n th e g e n e ra to rs in Tpc\ T c+1, th e g e n e ra to rs in Tpc are la rg e r th a n th e g e n e ra to rs in T a a n d th e g e n e ra to rs in T a are la rg e r th a n th e g e n e ra to rs in Tconj-
By T h e o re m 2.3.7 th e set { [y,x] \ x , y £ A c, w ( y ) = c , w ( x ) = 1 } g en era tes th e g ro u p 7c_t_1( G ) /7c+2( ( j) . T his m ean s th a t in m a k in g A'c c o n siste n t, we will find a set of re la tio n s th a t expresses each g e n e ra to r in T \ TC+ 1 as a w ord in T c+ i. It
is possible to d escrib e an alg o rith m , often called th e tails routine, th a t ex p resses each g e n e ra to r in Tconj as a w ord in Tpc. T h e n a m e com es fro m th e follow ing defin ition .
5.3.2 Definition Let (>1, re) be a w eighted n ilp o te n t p re s e n ta tio n a n d let c
b e th e h ig h e st w eight of a g e n e ra to r in A. If y is a re d u c e d w ord in A , w rite y as a p ro d u c t y \ y2 w ith w( y i ) < c a n d w ( y2) > c. T h e n y2 is called th e ta il of th e
w ord y. If y is th e rig h t h a n d side of a re la tio n x = y in 5 , th e n 7/2 is called the tail o f the relation x — y.
T h e ta ils ro u tin e is d escrib ed in th e n ex t sectio n . W e assu m e for now th a t all g e n e ra to rs in Tconj have been rep la ced by w ords in Tpc.
Let U be th e free a b e lia n g ro u p on th e g e n e ra tin g set T a U Tpc. R en am e th e g e n e ra to rs in th is lin early o rd ered set as a nc + i , . . . , a n , w h ere n c is th e n u m b e r o f g e n e ra to rs in A c. A p p ly in g T h e o re m 5.2.6 to th e w eig h ted n ilp o te n t p re s e n ta tio n re su lts in a set of red u ce d w ords in U Tpc. T h e u n io n of th is set a n d th e set { r ( x 1cr, . . . , x n a) \ r £ 71 } g e n e ra te s a su b g ro u p V of U. Let
I'c C {n c -f 1 , . . . , n c+i} a n d { = w a | i 6 / ' } b e a set of re la tio n s for U / V
w ith a re la tio n m a trix in row H erm ite n o rm al form .
Now let I"c+ i C I'c be th e indices of g e n e ra to rs for w hich m i > 1 a n d d e fine J c+i = { n c -f 1 , . . . , n c+1 } \I'C U Jc+i- All g e n e ra to rs in Ta a n d Tpc\ T c+1 can be ex p ressed as w ords in Tc+i by T h e o re m 2.3.7. T h e lin e a r o rd erin g on Tpc U Ta g u a ra n te e s th a t each of th ese g e n e ra to rs a p p e a rs w ith e x p o n e n t 1 on th e left h a n d side of a re la tio n for U/ V. T h is m e an s th a t th e set of g e n e ra to rs c o rre sp o n d in g to J c+ i is a su b se t of Tc+1. E lim in a tin g all o th e r g e n e ra to rs fro m th e q u o tie n t sy stem A'c we get a co n sisten t w eig h ted n ilp o te n t q u o tie n t sy stem A : + 1 = ( ( X , 7 l ) , ( A c+1, 5 c+1,iü c+1),o-c+1) re p re se n tin g th e q u o tie n t G /j c+2{G).