C OMPOSITIO M ATHEMATICA
F. B. C ANNONITO R. W. G ATTERDAM
The word problem in polycyclic groups is elementary
Compositio Mathematica, tome 27, n
o1 (1973), p. 39-45
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39
THE WORD PROBLEM IN POLYCYCLIC GROUPS IS ELEMENTARY
F. B. Cannonito and R. W. Gatterdam Noordhoff International Publishing
Printed in the Netherlands
1. Introduction
It is well known that the solution of the word
problem
forfinitely
pre- sented(henceforth ‘f.p.’)
groups can bearbitrarily
difncult in the sensethat it can have any
preassigned recursively
enumerabledegree (see
Boone
[2], [3],
andClapham [7]).
Thestudy
of the wordproblem
with respect to the finer
degrees
of the relativizedGrzegorczyk hierarchy
was
begun by
Cannonito[4]
and continuedby
Gatterdam[8].
A com-prehensive background
to this work can be found in the authors’ paper[5],
of which aknowledge
on the part of the reader is assumed. The exist-ence
of f.p.
groups with wordproblem stricly cn(A) computale
in the re-lativized
Grzegorczyk hierarchy, for n >_
4 andA,
arecursively
enumer-able subset of the natural
numbers,
was shownby
Gatterdam[9].
It shouldbe observed that the
degree
ofsolvability
of the wordproblem
for a fini-tely generated (henceforth ‘f.g.’)
group is an invariant of thef.g.
presen-tation.
Groups
with wordproblem 03B5n(A) computable
are said to be’03B5n(A)
standard’.The
question being
considered here is in theopposite
direction andstems from current efforts to
locate,
ifpossible,
with respect to theGrzegorczyk hierarchy,
the levelof computability
of the wordproblem
inf.p. residually
finite groups(see [6] for
somediscussion). Thus, knowing
how difncult the word
problem
can become forf.p.
groups, we ask in-stead
how ’easy’ is
it for classesof f.g.
groups with known solvable wordproblem?
Inparticular,
some classes have wordproblem
solvableby (Kalmâr) elementary functions,
03B53 in thehierarchy.
These classes include finite groups, free groups,automorphism
groupsof f.g.
free groups, braid groups andf.g.
abelian groups. Morerecently,
the authors in[6]
haveshown
why
one relator groups may not haveelementary
wordproblem.
The main result of this paper is to show the word
problem
forpolycyclic
groups
and, hence, f.g. nilpotent
groups iselementary. Now,
Auslanderhas shown in
[1 ] that
anypolycyclic
group is embeddable inGl(n, Z)
forsuitable n, and since
Gl(n, Z)
isisomorphic
to theautomorphism
group of the free group of rank n, we haveimmediately
from[5] Corollary
3.7that the word
problem
inpolycyclic
groups iselementary.
In this paper40
we derive this result
by analysing
the construction of apolycyclic
groupgiven by (finitely many) cyclic
extensions. In this manner, we also show thecomputability
level for a wider classof groups
iselementary
and de-monstrate a
computability
levelpreserving
construction.2.
Group
ExtensionsIn this section we
develop
some basic notionsconcerning
the compu-tability
of group extensions. Thebackground
material on extensions canbe found in MacLane
[11 ],
which we use without further reference.Consider an
extension, 0 ~ K ~ G ~ Q ~ 1
with both K andQ f.g.
and 03B53 standard. We write the group
operation
in K andQ additively
even
though they
ingeneral
will not be abelian. Adeparture
from theMacLane
approach
is that we willdistinguish
between K andx(K)
G.For every q E
Q
let2(q)
be alifting
into G(in general
not a homo-morphism)
so thatu2(q)
=1Q.
Thenconjugation
ofx(K) by 2(q)
in Gyields
ahomomorphism ik : Q ~
AutK/In
K.For q
EQ
let0(q)
e03C8(q)
be a choice of
automorphism
of K in the class03C8(q).
Since K isf.g.
and03B53
standard, 0(q)
is 03B53computable
forfixed q (see [5];
fork1,···, k"
generators of K and
w(k1, ···, kn)
eK, 0(q):w,
thatis,
theimage
of wunder
0(q),
can be obtained fromw(~(q):k1, ···, ~(q):kn) by
abounded minimalization and
w(~(q):k1, ···ü ~(q):kn)
can be ob-tained from
w(k1, ···, kn) by
an 03B53 limitedrecursion).
The group G is determined
(up
to naturalisomorphism) by
the0(q)
and a function
f : Q x Q ~ K
which expresses the deviation of 03BB, frombeing
ahomomorphism Q ~
Gby
The associative law in G
yields
Recalling ~(q):k
=GÀ(q)+k-À(q), conjugation by
the left andright
side of
(1)
inx(K)
Gyields
By taking 03BB(1)
= 0 we may as well have0(l)
=1 K
andf(q, 1)
=xOK = f(1, q) foraine Q.
Now a
simple computation ([11 ] lemma 8.1)
shows G ={(k, q) 1 k E K,
q e
Q}
as a set withmultiplication
and inversiongiven by
LEMMA:
If
under the above conditionsf : Q
xQ ~
K is f9’3computable
with respect to standard indices on
Q
and K then G is f9’4 standard.PROOF: We need f9’3
pairing
functionsJ, Ml, M2 (see [5]; Ml
is Kand
M2
is L of[5] § 2)
and f9’3 standard indices(il,
ml,jl)
and(iz,
m2,j2)
of K andQ respectively (again
see[5]).
Then G =
{(k, q)lk e K, q
eQI
as a set so we definei(G) by i(k, q) = 1(i1(k), i2(q)).
Thusx ~ i(G) ~ M1(x) ~ i1 (K) ^ M2(x) ~ i2(Q)
soi(G)
is f9’3 decidable.
By (4)
and(5) m : i(G)
x1(G) - i(G) and j : i(G) ~ i(G)
can be de-fined in terms of
f and 0 suitably
encoded. Thuslet j : i2(Q)
xi2(Q) --, il(K) by J(i2(p), i2(q)) = iif(p, q) and : i2(Q) i1(K) ~ i1(K) by
(i2(q), i1(k))
=i1(~(q):k).
We haveassumed f
is f9’3computable.
Postponing
thecomputability of ,
we can encodemultiplication
and in-version as follows:
Thus to show G is 03B54 we need to
prove e
is tff4computable.
It sufficesto show
e(x, y)
is tff4 for x Ei2(Q) and y
=il (a
generator ofK)
sincethen a standard 03B53 bounded recursion can be used to
compute (x, y)
for
arbitrary
y Ei1(K) (see [5 ] proof
of Theorem3.2). Now,
let x =i2(q0 ··· q03B2)
for qu - - - qp afreely
reducedword, each q03B1 being
a generatoror inverse of a generator of
Q.
Since K andQ
aref.g.
there arefinitely
many
values e(x, y)
for x= i2 (generator
ofQ
or the inverse of agenerator of
Q) and y
=il (generator
ofK) so (x, y)
is03B54
iffi
= 0.By
means of a recursion on
03B2,
let x’ -i2(q1 ··· q03B2)
and xo =i2(qo).
ThenSince this recursion has no a
priori
03B53bound,
the best that can be said ingeneral
isthat 4J
is 03B54computable and, hence,
G is an 03B54computable
group.
To show G is tff4 standard let a 1, ..., as
generate
and as+1, ···, as+t generateQ
so G isgenerated by
We must show the
homomorphism
i :a1, ···, as+t;) --+
Gby a03B1
H(aa , 0)
if 1 ~ a~ s, a03B1
H(1, aa)
if s + 1 ~ a ~ s + t, is tff4computable,
for then G is tff4 standard
by
Theorem 3.4 of[5].
The idea is to write afreely
reduced word in the free group asb0 ··· ble
for each bsymbol
agenerator a03B1
or an inversea03B1-1.
Then if03C4(b1 ··· bfJ) = (k’, q’)
we have42
Thus the
encoding
of i can begiven by
a recursioninvolving
e3 com-putable
functions.For the interested reader we record the details of the above. The no-
tation is that of
[5] (see
also[10]
p. 222ff.)
as is the assumedencoding
ofthe free group. Define
Then
We note that if K and
Q
aref.g. e"(A)
standardfor n ~
3 then Gis
f.g. e""(A)
standardby
the sameproof, replacing
03B53by 03B5n(A)
and03B54
by 03B5n+1(A).
Of more interest are situations when the recursion involved in thecompitability
of î can be boundedby
an 03B53 functionso that G is 03B53 standard.
COROLLARY:
If K is.f’.g.,
03B53 standard andQ
isa f.g. free
group then G is 03B53 standard.PROOF: In this case the extension
splits, i.e.,
thelifting
03BB can be takento be a
monomorphism. Then f(p, q)
= 0 for all p, q EQ and 0 : Q -
Aut k is a
homomorphism.
The recursion
defining
î becomesTo show G is 03B53 standard we must show i is 03B53
computable.
It sufficesto show
Ml
t andM2 î
are03B53 computable.
SinceM2 i
is theencoding
of ahomomorphism
from af.g.
free group to the 03B53 standard groupQ, it
is 03B53
computable.
To see
Ml t
is d"computable
first observe thatp(x)
encodes eithera generator or inverse of a generator. Now for u the index of a generator
or inverse thereof in
Q,
and w Ei1(K), (u, w)
is C3computable
since Kis
f.g.
and o3 standard. We must show the recursionyielding Ml î
is C3limited. For w E
i1(K)
let«w)
denote thelength
of the word(relative
tothe standard index
presentation)
w encodes.Similarly
for xencoding
aword in F =
a1, ···, as+t>;
let03B2(x)
be thelength
of the word. It sufficesto find an C3 bound on
CM1 -r
sinceM1(x) 03C103B6M1 (x)
exp«Ml (x).
J(s, 2)).
For uencoding
a generator or inverse thereof ofQ
andv encoding
a generator of K set z = max
03B6(u, v).
Then forw ~ i1(K), 03B6(u, w) ~
z -03B6(w).
Now in the recursiondefining Ml
î either mlor
isapplied.
Inthe first case
(Ml î
is boundedby
itsprevious
valueplus
1 and the latterby
z times itsprevious
value. ThusCM1 (x) ~
z exp03B2(x),
an 03B53 func-tion.
COROLLARY:
If K is f.g.
C3 standard andQ
isfinite
then G is C3 stan-dard.
PROOF:
Q being
finite it is C3standard, 1
is C3computable (there being only finitely
many values for theargument), and
is d-’computable (K being f.g.
and C3standard).
Theproblem,
asabove,
is to C3 limit03B6M1 .
Here let z = max03B6(u, v) + max (u, w),
the maximums takenover all u, w E
i2(Q)
and vencoding
a generator of K. Then(Ml (x) ~
z exp
03B2(x).
Observe that in the corollaries if K is assumed
f.g.
andcn(A)
standardfor n >_
3 the conclusion is that G isen (A)
standard.3. Main Results We are now
ready
to prove theTHEOREM:
Polycyclic
groups haveelementarily
decidableword problem,
i.e., are C3 standard.PROOF: If G is a
polycyclic
group it isnecessarily f.g.
and there existsa sequence
G0, ···, Gn
= G of groups such thatGo is cyclic (hence
C3standard)
andG., 1
is an extension ofG. by
acyclic
group for m n.Inductively
assumeGm
is C3 standard. Then it is extended eitherby
aninfinite
cyclic
group,i.e.,
a free group, or a finitecyclic
group.By
thecorollaries of the
preceding
sectionGm+1
is C3 standard.The theorem above can be
generalized
of course. LetG0
be the classof finite groups union that of
f.g.
free groups. DefineGn+1
to be theclass of extensions of a group in
gn by
a group inG0
and G =Un= 0 e. - By
the above argument the groups in G all aref.g.
and C3 standard. Wecan say more.
44
THEOREM: Let K be
aig. d"(A)
standard groupfor
n ~ 3 andQ
E td.Then any extension
of K by Q is f.g.
ande"(A)
standard.PROOF: K and
Q
aref.g.
so an extension G of .Kby Q
isf.g.
LetQ
EGm.
If m =
0,
then G is C3 standardby
the corollaries.Inductively
if m > 0 thenQ
is an extension ofQl
EGm-1 by Qo
etdo.
ConsiderHere
xo N
= ker pu G and G is an extension of Nby Qo E G0.
Wemust show N is
f.g.
andcn(A)
standard. Consider the naturalembedding N/K - G/K ~ Q.
Since03C3003BA0(N)
=0,
theimage
ofN/K
is ini (Q 1 ).
Thus we have an
embedding ( : N/K ~ Q1.
Nowif q
EQ,
there exists g E G such thatQ(g)
=i(q).
Thenpu(g)
=03C1(q)
= 0implies g
E03BA0(N)
so (
is onto. ThereforeN/K ~ 61
so N is an extension of thef.g. 03B5n(A)
standard group
K by Q E 9.
andby
induction isf.g.
anden (A)
stan-dard.
It should be noted that 9 contains
f.g.
abelian andpolycyclic
groupsso in
particular
an extension of af. g. 03B5n(A)
standard groupby
af. g.
abelian or
polycyclic
group preserves thecomputability
level.REFERENCES
[1] L. AUSLANDER: On a problem of Phillip Hall. Annals of Math., 86 (1967), 112-116.
[2] W. W. BOONE: Word problems and recursively enumerable degrees of unsolva- bility. Annals of Math., 83 (1966), 520-591.
[3] W. W. BOONE: Word problems and recursively enumerable degrees of unsolv- ability. A sequel on finitely presented groups. Annals of Math., 84 (1966), 49-84.
[4] F. B. CANNONITO: Hierarchies of computable groups and the word problem.
J. Symbolic Logic, 31 (1966), 376-392.
[5] F. B. CANNONITO and R. W. GATTERDAM: The computability of group construc-
tions, Part I. Word Problems, Boone, Cannonito, Lyndon eds., North-Holland Publishing Company, Amsterdam (1973).
[6] F. B. CANNONITO and R. W. GATTERDAM: The word problem and power problem
in 1-relator groups is primitive recursive (submitted for publication).
[7] C. R. J. CLAPHAM: Finitely presented groups with word problem of arbitrary degree of insolubility. Proc. London Math. Soc., 14 (1964), 633-676.
[8] R. W. GATTERDAM: Embeddings of primitive recursive computable groups (sub- mitted for publication).
[9] R. W. GATTERDAM: The computability of group constructions, Part II Bull.
Australian Math. Soc., 8 (1973) 27-60.
[10] S. C. KLEENE: Introduction to Metamathematics. Van Nostrand, Princeton, New Jersey (1952).
[11] S. MACLANE: Homology. Springer-Verlag, Berlin (1963).
(Oblatum 7-VrII-1972) Department of Mathematics
University of California - Irvine USA and
Department of Mathematics
University of Wisconsin - Parkside 1 USA
1 Support for the first author by the Air Force Ofhce of Scientific Research Grant AF-AFOSR 1321-67 and for the second author by the Wisconsin Alumni Research Fund Grant 130399 is gratefully acknowledged.