We assum e th e n o ta tio n of th e p rev io u s sectio n . In o rd e r to a p p ly th e co nsid e ra tio n s of th e p rev io u s section to checking th e co n sisten cy of a polycyclic p re se n ta tio n , let
G
be given by a polycyclic p re s e n ta tio n on th e g e n e ra to rs { a l 5 .. . ,an}
for i 6 I
for 1 <
i < j
<n
for 1 <
i < j
< n,i
0I.
W e assum e th a t th e rig h t h a n d side of each re la tio n is a red u ce d w ord. Let
N
be th e g ro u p defined by th e p re s e n ta tio n on th e set { a 2, . . . , a n } of g e n e ra to rs an dfor
i
£ I \{ 1 } for 2 <i
<j
<n
for 2 <
i <
j< n , i $ I .
A ssum e th a t th is p re s e n ta tio n for
N
is co n siste n t. T h e su b g ro u p ofG
g e n e ra te d by { a 2, . . . , a n } is a ho m o m o rp h ic im age ofN.
D efine a m a p ip: { a 2, . . . , a n } —■»N
as m a p p in g a; to.
If 1 ^ / , also defineip
: { a 2, . . . , a n } —>N
m a p p in g Gq to Fori
£ / \ { 1 } letqa
be th e te rm th e re la tio n s m, =wu
— 1 “ E d " a i a j U i = w i;j aid j a //1 = w~ + w ith re la tio n s m, =wa
a ~ l aj di — ai aj a~ 1 = w~3+™ii ( a i+ l i p , . . . , a nip) 1( a l (p)Tni, for 2 <
i
<j
<n
letqtJ
be th e te rma n d , ex te n d in g th e n o tio n of (^-closure fro m free g ro u ps to fa c to r g ro u p s of free g ro u p s, let S be th e (^-closure of th e n o rm al closure of
{ qa I i e A { 1 } } U { qij I 2 < i < j < n }.
By th e co n sid era tio n s of th e p rev io u s sectio n , tp in d u ces a m a p a t S >—> ( a ^ ) 5 on N / S w hich defines an en d o m o rp h ism of N / S. Since N is given by a co n sisten t polycyclic p re s e n ta tio n , th e n o rm a l form s of th e qij can be co m p u te d . N ote th a t th e re is a p o sitiv e in te g e r L such th a t = b eca u se th e re are no in fin itely ascendin g chains in th e polycyclic g ro u p N .
If 1 ^ I , th e n for 2 < i < n let r{ be (ai'^)ipa~ 1. T h e ip-clo su re T of th e n o rm a l closure of S U { rq | 2 < i < n } is th e sm allest n o rm a l su b g ro u p of N such th a t ip induces an a u to m o rp h is m on N / T w ith in v erse -0.
If 1 6 / , th e n th e re is a re la tio n a™1 = u>n . For 2 < i < n let be (ai(prrll)w^11 a ~ J w n . T h e n th e -closure T of th e n o rm a l closure of th e set 5 U { r { | 2 < z < n } U } is th e sm allest n o rm a l su b g ro u p of N such th a t ip in d u ces an a u to m o rp h ism on N / T such th a t its m j - t h pow er is th e in n e r a u to m o rp h is m of N defined v ia co n ju g a tio n by wn .
T h e m a p ip defines an a u to m o rp h is m ( th a t , if 1 E J , also satisfies th e co n d i tio n s of L em m a 3.2.2) on N in b o th cases if a n d only if th e su b g ro u p T is triv ia l in b o th cases.
For 2 < j < n we have th a t ajip = a n d , if 1 ^ I , a,ji/> = . By th e first p o stp o n in g p rin cip le a n d th e fact th a t w(a,2, • . . , a n ) is a red u ce d w ord, th e w ord w ( a 2, . .. , a n )a i collects to th e sam e n o rm a l fo rm as th e w ord
a 1w ( a 2<Pi. . . , cinp)- Now T is triv ia l if a n d only if th e follow ing e q u a tio n s are satisfied as eq u a tio n s in N TTl j a j a l = a 7 i _ 1 l a J a i ] for 2 < j < 71, j e I , [<7j Q-l | = [ a f c a j J a j for 2 < j < k < 7 i , m 1 aj a1 = [ a j a i J a T 1 1 - 1 for 2 < j < 71, i e l , m i a i d j = a l m i d \ if 1 6 1 , aj for 2 < j < 71, i i i .
T h is can be checked by collecting b o th sides of th e e q u a tio n to n o rm a l form b ecau se N is defined by a co n sisten t polycyclic p re s e n ta tio n . By in d u c tio n we get th e follow ing th e o re m .
3.4.1 Theorem (C o n sisten cy C heck)
A polycyclic p re s e n ta tio n on th e g e n e ra tin g set { a 1, . . . , a n } is co n sisten t if a n d only if th e follow ing w ords collect to th e e m p ty w ord