DIMACS Series in Discrete Mathematics
and Theoretical Computer Science
Volume 00, 19xx
Experimenting and computing with infinite groups
Gilbert Baumslag and Charles F. Miller III
Abstract. We describe an experimental approach to studying infinite groups
using a software package called Magnus being developed by the New York
Group Theory Cooperative. This approach emphasises infinite groups and
partial and experimental computation. These computations are frequently
inconclusive and may only occasionally succeed. Such experimentation can
guide theoretical development and lead to new and interesting questions.
1. Introduction
We want to describe here an experimental approach to studying infinite groups,
particularly finitely presented groups. The ongoing software project, called Magnus, which is being undertaken by the New York Group Theory Cooperative, together with a number of mathematicians around the world, has been designed to
make such an approach possible. The Magnus project is not intended to compete
with the highly successful Gap and Cayley packages. Its emphasis is on infinite
groups, partial and experimental computations and easy access to existing packages.
Magnus is comprised of a back-end, a session manager and a front-end. The
back-end consists, for the most part, of a number of self-contained algorithms. The
front-end provides the end-user with an easy-to-use graphical user-interface. The
session-manager, in particular, communicates between the front-end and the backend, enabling the system to keep track of any information that has been obtained
as well as to post various log messages. It is convenient to think of Magnus as
a group-theoretical self-service center, equipped with a large number of diagnostic
and other tools. It is these tools and the way that they are made available through
the front-end, that make possible the design and implementation of experiments
with potentially infinite groups, their elements and the morphisms between them.
Two windows are provided to make the center easy to use. The first window logs
a running commentary of a session with Magnus . The second, the workspace,
furnishes the end-user with space to work in. The workspace contains a checkin facility for entering abelian groups, free groups, nilpotent groups, metric small
cancellation groups, one-relator groups, words, morphisms and so on, into it. The
1991 Mathematics Subject Classification. Primary 20.
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c
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American Mathematical Society
1052-1798/00 $1.00 + $.25 per page
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GILBERT BAUMSLAG AND CHARLES F. MILLER III
end-user starts a session by checking a group into the workspace, using one of the
items provided by the check-in menu. Algebraic objects, such as groups, words and
the like, are represented in the workspace by icons as are problems connected to
them. A panel of control buttons allows one to model in Magnus the way in which a
mathematician might try to solve a particular problem or carry out an experiment.
Additionally, computations or problems can be suspended, resumed, terminated or
canceled. A complicated, possibly time-consuming computation, can be controlled
by the end-user by allocating Abstract Resource Credits, usually simply seconds,
to it. Many such complex computations are managed by Computation Managers,
which can also manage other Computation Managers. This makes it possible to
communicate the results of any computation to the system as a whole and allow
also for their use in other computations. Information about a particular group, for
example, is gathered together in its group information center, thereby making it
accessible to the user.
Since most problems are recursively unsolvable (see below) we can, at best,
enumerate all instances of a certain phenomena or look for instances by randomly
trying possibilities (partial recursive procedures or computations). Magnus takes
this situation into account. It also provides some of the tools needed for getting
insights into what to try to prove about infinite groups. Combinatorial group
theorists have often calculated with examples by hand. But now computing devices
have become fast enough to give hope of computing examples of significantly greater
complexity than one can expect to do by hand.
There are some inherent problems with using software such as that provided by
Magnus. Experiments only provide guidance for theoretical development. Even
the best software cannot prove theorems, nor can such software concoct definitive examples. Experiments are often inconclusive and carrying them out can be
extremely frustrating. Computational tools can often only implement partial or
random algorithms and are limited by severe size restrictions. However they can
also provide a starting point for investigating even the most intractable of problems
and consequently can be of considerable help.
It is worth-while to explain why experimenting with infinite groups is of necessity ad hoc. Such an explanation is rooted in an historic paper in 1912, in which M.
Dehn explicitly raised the following three problems about finitely presented groups.
The word problem: Let G be a group given by a finite presentation
G = hX; Ri.
Is there an algorithm which decides whether or not any given X-word, that
is a word in the given generators X of G, is equal to the identity?
The conjugacy problem: Let G be a group given by a finite presentation
G = hX; Ri.
Is there an algorithm which decides whether or not an arbitrary pair of
X-words v and w are conjugate in G?
The isomorphism problem: Let
Y = {y1 , y2 , . . . , . . . }
be a countably infinite set and let F be the recursive set of all finite presentations whose generators are finite subsets of Y . Is there an algorithm
EXPERIMENTING AND COMPUTING WITH INFINITE GROUPS
3
which determines whether or not an arbitrary pair of such presentations in
F define isomorphic groups?
Some 20 years after Dehn proposed these problems W. Magnus, 1930, proved
his famous Freiheitssatz.
Theorem 1.1. Let G be a group with a single defining relator, i.e.,
G = hx1 , . . . , xq ; ri.
Suppose r is cyclically reduced, i.e., the first letter in r is not the inverse of the last.
If each of x1 , . . . , xq actually appears in r, then any proper subset of {x1 , . . . , xq }
freely generates a free group.
This led to the first general theorem about groups with solvable word problems,
which is due also to W. Magnus, 1932.
Theorem 1.2. The word problem for a 1-relator group has a positive solution.
Somewhat surprisingly, Theorem 2 turned out to be atypical. In fact almost
50 years after Dehn put forward the three problems above they were answered in
the negative. The breakthrough was made by P.S. Novikov in 1954.
Theorem 1.3. There exists a finitely presented group with an insoluble word
problem.
Notice that this implies that there is a finitely presented group with an unsolvable conjugacy problem.
Theorem 3 provided the means for Adian, in 1957, and Rabin, in 1958, to prove
a most striking negative theorem about finitely presented groups which depends,
for its formulation, on the notion of a Markov property.
Definition 1.4. An algebraic property (i.e. one preserved under isomorphism) of finitely presented groups is termed a Markov property if
1. there exists a finitely presented group with the property,
2. there exists a finitely presented group which cannot be embedded in, i.e. is
not isomorphic to a subgroup of, a group with the property.
Theorem 1.5. Let M be a Markov property. Then there is no algorithm which
decides whether or not any finitely presented group has this property M.
The wide applicability of Theorem 4 is illustrated by noting that the following
are Markov properties:
1. triviality;
2. finiteness;
3. abelianess;
4. having solvable word problem;
5. simplicity;
6. freeness.
It is not clear that simplicity is a Markov property, but it is.
It is an easy matter to deduce from Theorem 4 that if G is a group defined by
a given finite presentation, then there is no algorithm which decides whether or not
any finite presentation defines a group isomorphic to G.
A similar, much simpler, theorem about subgroups of a group, due to G. Baumslag, W.W. Boone and B.H. Neumann followed shortly afterwards in 1959, as well
as one about elements.
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GILBERT BAUMSLAG AND CHARLES F. MILLER III
Theorem 1.6. Let P be an algebraic property of groups. Suppose that
1. there is a finitely presented group that has P;
2. there exists an integer n such that no free group of rank r has P if r ≥ n.
Then there is a finitely presented group G such that there is no algorithm which
determines whether or not any finite set of elements of G generates a subgroup with
P.
It follows, in particular, that there is a finitely presented group G with the property that there is no algorithm which decides whether or not any finitely generated
subgroup of G is finitely presented.
Theorem 1.7. There is a finitely presented group G0 such that there is no
effective procedure which determines whether or not a word in the generators of G0
represents
1. an element in the center of G0 ;
2. an element permutable with a given element of G0 ;
3. an n-th power with n > 1 an integer;
4. an element whose conjugacy class is finite;
5. an element of a given subgroup of G0 ;
6. a commutator, i.e., of the form x−1 y −1 xy;
7. an element of finite order.
These results demonstrate that from an algorithmic standpoint, finitely presented groups constitute a somewhat intractable class of groups. The extent of this
algorithmic intractability was underlined by a remarkable theorem of G. Higman,
proved in 1961.
Theorem 1.8. Let G be a finitely generated group. Then G is a subgroup of a
finitely presented group if and only if G can be presented in the following form
G = hx1 , . . . , xq ; r1 , r2 , . . . i
(q < ∞)
where r1 , r2 , . . . is a recursively enumerable set of defining relations.
It follows from Higman’s theorem that finitely presented groups with all sorts
of undecidability features, such as unsolvable word problems, are not uncommon.
For an extensive survey of decision problems see [16].
Despite the fact that any systematic approach to almost any problem about
finitely presented groups is bound to fail, experimentation in some particular instances can be helpful. We detail below, some general as well as some specific
scenarios for dealing with a few well-known and other problems about finitely presented groups. Each of these scenarios impacts on and has meaning for the others.
2. The isomorphism problem
We have already noted that if G is the group defined by the finite presentation
< X; R >, then there is no algorithm which decides whether or not any finite
presentation < Y ; S > also defines G. In practice, one is often confronted not
by an algorithmic problem, but a simple question: given two finite presentations
< X; R > and < Y ; S >, do they define isomorphic groups? If we denote the
group presented by < X; R > by G and that presented by < Y ; S > by H, then
we are asking whether G is isomorphic to H. Now it follows from a theorem of
Tietze that we can enumerate all of the finite presentations of G and all of those
EXPERIMENTING AND COMPUTING WITH INFINITE GROUPS
5
of H. Consequently, if G and H are isomorphic, given enough time and space,
we will eventually find this out (granted that we have coded a procedure to carry
out such enumerations). Of course, if G and H are not isomorphic, this procedure
will not tell us anything. There are, however, other things that we can try. For
instance, we can, in many cases only to a limited extent, check whether G and
H have the same abelianisations, whether they have the same nilpotent quotients,
whether they have the same metabelian quotients, whether they have the same
finite images, whether they have same number of subgroups of a given finite index,
whether they have the same integral homology groups, whether they have finite
rewriting systems, whether they have automatic structures, whether their affine
algebraic sets of representations into SL2 (C) have the same dimensions, whether
they have the same elements of finite order and so on. If we are able successfully
to obtain some conflicting information, then we will know that G and H are not
isomorphic. The way, as well as the order, in which we carry out these tasks can be
viewed as the design and implementation of the kind of experiment about G and H
that we have been alluding to above. There are many variations on such a design.
For example, we might try to compute the set Hom(G, T ) of all homomorphisms
of G into some nice target group T for 10 seconds and then try to compute the
corresponding set Hom(H, T ), also for 10 seconds. If the cardinality of these two
sets is vastly different, this would suggest that G and H are not isomorphic. We
could also put similar time constraints on some of the procedures above or any new
ones we can devise, which might give us some indication as to whether G and H are
isomorphic. At present Magnus does not provide access to all of these procedures.
However we hope that, in due course, it will provide access to all of them.
3. The conjugacy problem
We discuss next, in a little more detail, another instance of this approach.
Suppose that we are given a finite presentation
G =< x1 , . . . , xn | r1 = 1, . . . , rm = 1 >
and that we are interested in determining whether or not two X-words u and v are
conjugate in G, where X = {x1 , . . . , xn }. Such a problem might arise for instance
if u and v represent two closed loops in a finite complex and we want to know
whether they are freely homotopic.
Now if the conjugacy problem for G is unsolvable, then there is no algorithm
which decides whether or not any pair of X-words represent conjugate elements of
G. The situation is different if we are given just the two words u and v. ¿From
an algorithmic standpoint, the problem as to whether u and v represent conjugate
elements of G is solvable. The only difficulty is that we do not know what the
answer is! It is worth-while to examine what it is that is involved. To this end, let
R denote the normal closure in the free group F on X of the elements r1 , . . . , rm .
If we identify G with F/R, then u is conjugate to v in G if and only if there exists
an X-word c0 and an X-word r0 ∈ R such that
u = x−1
0 vx0 r0 .
Let H0 be the subgroup generated by r1 , . . . , rm , H1 the subgroup generated by
H0 together with the conjugates of its generators by x±1
i , and so on. It follows that
R can be expressed as an ascending union of these finitely generated subgroups
Hi (i = 0, 1, ...) and hence is recursively enumerable. From our standpoint, we
6
GILBERT BAUMSLAG AND CHARLES F. MILLER III
would like to view this sequence of finitely generated subgroups as better and better
approximations to R, the larger the value of i, the better the approximation. We
can enumerate all products of the form x−1 vxr, where x ranges over F and r ranges
over R. Now given a finitely generated subgroup H of a finitely generated free group
F freely generated by X there is a finite state automaton A over X such that H
comprises the language of A. This implies that there is an algorithm for determining
membership in H. In particular, there is for each i, a finite state automaton Ai
which decides membership in Hi . More generally, there is, for each i an analogous
automaton which decides whether u is conjugate to v, modulo Hi , i.e., whether
there is an X-word x such that x−1 ux ∈ vHi . If there is such a word x, then u and
v are conjugate in G by x. If this is not the case and if n is reasonably large, then
this suggests that u and v are not conjugate in G. Magnus provides the tools to
carry out this procedure, once again if time and space permit. The discussion of
the isomorphism problem above, indicates that further experimentation is possible.
One can check whether u and v represent elements of G which perhaps are not
conjugate modulo some term of the lower central series of G, or modulo the second
derived group of G or modulo some normal subgroup of finite index in G, or,
more generally, in some homomorphic image of G. We might try to find a finite
rewriting system for G, or an automatic structure or a bi-automatic structure and,
if we are successful, use the additional knowledge about G to launch some further
experiments. Magnus at present provides some of these options. It will eventually
be more fully equipped. The point of the matter is that although these computer
experiments may be inconclusive, an examination of the results of such experiments
may provide some guidance, leading the researcher to obtain definitive answers by
sifting through the evidence collected.
4. Hyperbolic and small cancellation groups
Despite the fact that almost all problems about finitely presented groups are
algorithmically unsolvable, there are many classes of finitely presented groups for
which a rich algorithmic theory does exist. In general, given a finitely presented
group, one does not know whether or not it lies in one of these good classes.
Nonetheless, if one assumes, without prior knowledge, that it does and simply begins to apply various algorithms to it, something positive sometimes emerges. This
is true of the the well-known Knuth-Bendix procedure, which looks for a so-called
finite, confluent rewriting system for a given finitely presented group, whether it
has one or not. The method of coset enumeration of Todd-Coxeter is another such
procedure which can be applied without prior knowledge of any likelihood of success. These methods are described in the fundamental book of C.C. Sims and play
an important role in computational group theory. Their role has been well documented and we will not concern ourselves with them here. Our objective at this
juncture is to illustrate how one might be able to use knowledge of a well-behaved
class of groups to help carry out computations in general. We will therefore turn
our attention to a discussion of hyperbolic groups, introduced and studied first by
M. Gromov in his remarkable paper [8].
In order to define a hyperbolic group, we need first to recall the definition of
the Cayley graph Γ = Γ(G) = ΓX (G) of a group G relative to a set X of generators
of G. Γ is a directed graph with vertex set G and edges all triples (g, a, ga), where
g ∈ G and a ∈ A = {X ∪ X −1 }. We respectively term g the origin and h the
EXPERIMENTING AND COMPUTING WITH INFINITE GROUPS
7
terminus of the edge (g, a, h). The origin and terminus of an edge are referred
to as its extremities. We term a sequence γ of (not necessarily distinct) vertices
g0 , . . . , gn of Γ a path of length n if either n = 0 or, in the case where n > 0, if,
for each i = 0, . . . , n − 1, either gi = gi+1 or there exists an edge whose origin is
gi and whose terminus is gi+1 . We term g0 the origin and gn the terminus of γ
and refer to them as the extremities of γ and we say that γ goes from g0 to gn .
Since (ga, a−1 , g) goes from ga to g and X generates G, it follows that he Cayley
graph is always path connected. If P and Q are vertices in a Cayley graph ΓX ,
then the distance d(P, Q) between them is defined to be the minimum length of a
path from P to Q. For each path that goes from P to Q, there is a corresponding
path that goes from Q to P . It follows that Γ is a metric space. A shortest
path from the group element g to the group element h is termed a geodesic and
a “triangle” in ΓX (G) is termed a geodesic triangle if its sides are geodesics. M.
Gromov [8] has termed a geodesic triangle δ − thin if every point on one side of
the triangle is no further than δ from at least one point on one of the other two
sides, i.e., each side of the triangle is contained in a δ-neighborhood of the union of
the other two sides. The group G is then termed hyperbolic if there exists a δ such
that every geodesic triangle in ΓX (G) is δ-thin. Hyperbolic groups have solvable
word, conjugacy and isomorphism problems. Thus they constitute a good class of
groups to model finitely presented groups on. Indeed, M. Gromov has proved that
if one ”counts” in a suitable way, then ”most” finite presentations define hyperbolic
groups.
In order to explain how this point of view can be exploited experimentally, we
need to describe what is now known as Dehn’s algorithm (because it was used by
Dehn to solve the word problem for the fundamental groups of orientable surfaces
in 1912). Suppose that a group G has a finite presentation
G = hX; Ri
which is symmetrized, i.e., R is closed under inverses and cyclic permutations.
Suppose that for each reduced X-word w representing the identity in G there exists
r = a1 . . . al ∈ R such that
w = b1 . . . bm a1 . . . an c1 . . . cs
where
l
,
2
i.e., w contains more than half a relator (here bi , aj , ck ∈ X ∪ X −1 ). We then say
that the given presentation is a Dehn presentation of G. We can solve the word
problem for G as follows. Given any reduced X-word w, by inspecting the finitely
many elements of R, we can determine whether or not more than half of one of
them is a subword of w. If not, then w 6= 1. Otherwise, w = tuv where u is more
than one half of r = us ∈ R. So
n >
w =G ts−1 v
Now the the length of the reduced X-word w1 representing w is smaller than
the length of w. So we can repeat the process with w1 . Inductively we can therefore
determine whether or not w = 1 in G. Lysionuk and Shapiro, independently, proved
that a group G is hyperbolic if and only if it has a Dehn presentation.
One might ask what relevance this has to an arbitrarily given finite presentation
of a some group G. Given a reduced X-word w, one simply goes ahead and applies
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GILBERT BAUMSLAG AND CHARLES F. MILLER III
Dehn’s algorithm. Even if this presentation is not of the correct kind, one may
nevertheless find that Dehn’s algorithm leads to the conclusion that w =G 1. Of
course the procedure may well terminate inconclusively with a word of positive
length and we are left without knowing whether or not w =G 1.
The solution of the conjugacy problem for hyperbolic groups is not as easy to
describe. It suffices here to say that it does provide a model for experimentation
for finitely presented groups as a whole.
There is no algorithm whereby one can decide whether or not any finitely
presented group is hyperbolic. There is, however, an important class of groups, the
so-called one-sixth groups which are hyperbolic. By definition, a group G is termed
a one-sixth group if it has a finite, symmetrised presentation
G = hX; Ri
which satisfies the following condition: if r, s ∈ R and if either more than one sixth of
the length of s cancels on computing the reduced word representing rs or more than
one sixth the length of r cancels on computing the reduced word representing rs,
then r = s−1 . Such presentations turn out, by a theorem of M. Greendlinger (1960),
to be Dehn presentations. Therefore, by the theorem of Lisionok and Shapiro, these
groups are hyperbolic. The one-sixth condition is readily checked and so provides
a means for solving the word problem in a select collection of finitely presented
groups. Dehn’s algorithm is a particularly useful one. A similar algorithm exists
also for solving the conjugacy problem, but it can be very space consuming. As
we have already noted, these algorithms provide us with tools to deal with the
word and conjugacy problems for arbitrary finitely presented groups. Somewhat
surprisingly such tools can lead to successful conclusions.
5. Some classes of groups with good algorithmic properties
Many of the tools in Magnus depend on the fact that various classes of groups
enjoy a number of good algorithmic properties. We have described one such class,
the class of hyperbolic groups. Here we want to discuss some additional wellbehaved classes. One of the most well-known of these is the class of finitely generated abelian groups. The Smith normal form algorithm for integer matrices makes
it possible to determine their structure from an abelian group presentation. Thus,
for example, if we want to determine if a word in some finitely presented group does
not define the identity, we check to see if it does not define the identity modulo
its commutator subgroup. If we want to determine whether two finitely presented
groups are not isomorphic, we check to see if their abelianisations are not isomorphic, and so on. Such checks are, of course, often inconclusive. But they, and a
number of variations and generalised versions of them, sometimes provide important
information.
The word, conjugacy and isomorphism problems for finitely generated nilpotent groups, indeed for polycyclic groups as well, are all algorithmically solvable.
These tests can similarly be carried over to tests for the word, conjugacy and isomorphism problems for finitely presented groups as a whole. As an illustration of
how this might be useful, consider a hypothetical situation in which we are given
two subgroups H and K of a finitely presented group G. Our objective is to prove
that the intersection H ∩ K is not finitely generated. So we compute the minimal
number d(c) of generators of their intersection modulo the c − th term of the lower
EXPERIMENTING AND COMPUTING WITH INFINITE GROUPS
9
central series of G. Such a computation. although often impractical, is theoretically
possible. Suppose that we find that the sequence
d(1), d(2), . . . , d(c), ...
grows rapidly. This suggests that H ∩ K is not finitely generated. This experiment
would be of little use if we cannot use the data we have obtained to help us to actually prove that the intersection of H and K is not finitely generated. Additionally,
it makes sense also to look for suitable target group T and a homomorphism of G
into T where we are able to prove that the image of H ∩ K is not finitely generated.
This involves looking for random homomorphisms of G into T . Such experiments
are mainly impossible to carry out by hand, but at they are sometimes feasible to
conduct with computers.
The class of free groups is an especially useful one. Most problems about free
groups are algorithmically solvable. The solutions of many of them can be applied to
provide partial results about finitely presented groups. We have already described
one such application to the conjugacy problem. Nielsen’s algorithms for finding free
bases for finitely generated subgroups of a free groups and deciding membership
in such finitely generated subgroups, are two particularly useful ones. The latter
algorithm carries over to a procedure for solving the membership problem in any
finitely presented group, using the method described at the outset concerning the
conjugacy problem. Whitehead’s algorithms for reducing the length of words and
for recognizing primitive elements, i.e., elements which can be made part of a free
basis, are additional powerful and important algorithms which are likely to have
wide experimental use.
One-relator groups are another important class of groups. In particular, they
help to focus on two of the most widely used constructions in the study of finitely
presented groups - HNN extensions and amalgamated products. As already pointed
out earlier, one-relator groups have solvable word problems. The conjugacy problem
and isomorphism problem for such groups are still unresolved, despite intensive
study. Their investigation is a fertile field for computational methods. We will
have a little more to say about this when we discuss some open problems where
experimental computational methods might be helpful, in due course.
Our next objective is to describe some case histories, where a little experimentation produced a few unexpected results. Magnus is only now becoming available
so many of our examples involve either the the use of previously available software
or of preliminary versions of Magnus. Our first case history concerns work of C.F.
Miller and P.E. Schupp.
6. Balanced presentations
A finite presentation is termed balanced if it has the same number of generators
as relations. There is an old question of Andrews and Curtis concerning balanced
presentations of the trivial group, which is still open. In the late 1970’s, Miller and
Schupp observed that
< a, b; a−1 bn a = bn+1 , b−1 am b = am+1 >
where m, n > 0, is a presentation of the trivial group. This led them to ask when
presentations of the form
< a, b; a−1 bn a = bn+1 , w = 1 >
10
GILBERT BAUMSLAG AND CHARLES F. MILLER III
define the trivial group, where n > 1 and w is a word on a and b. An obvious
necessary condition is that the exponent sum on the letter a in the word w be ±1,
i.e., that the groups involved be perfect. Lacking any insight into the question,
they turned to using a coset enumeration package developed by George Havas.
As a pseudo-test of whether such groups are always trivial, they enumerated the
cosets of the cyclic subgroup generated by one of the generators, say a. The simplest
examples quickly yielded to the computer’s power, which showed that that they were
all trivial. More complicated examples quickly exceeded the computer’s memory
(which was small by present standards). With the help of Havas and Mike Newman,
they tried increasingly larger and more complicated words w on increasingly larger
computers. In every case, each of the groups involved turned out to be trivial.
Eventually, armed with huge amounts of experimental evidence, they found the
right insight to prove this condition is also sufficient. In fact, they were able to
prove somewhat more, namely,
Theorem 6.1. Let G be a perfect group generated by two elements a, b which
satisfy the relation a−1 bp a = bq , where p, q > 0 are relatively prime. Then G is the
trivial group.
7. Homology
In a talk at the 1992 Victorian Algebra Conference in Melbourne, Grant Cairns
asked whether the ranks of the integral homology groups of a finitely generated
nilpotent group obeyed a “one-mountain” principle. By way of explanation, it was
known that if h is the Hirsch length of a finitely generated nilpotent group and if
rk A denotes the torsion-free rank of an abelian group A, then
rk H0 (G, Z) = rk Hh (G, Z) = 1,
rk Hi (G, Z) ≥ 2, f or i = 1, . . . , h − 1
and
rk Hi (G, Z) = zrk Hh−i (G, Z)
for i = 0, . . . , h. So the torsion-free ranks are symmetric about the “middle dimension” h/2. Cairns asked whether whether they are monotone increasing up to
the middle dimension (and thus monotone decreasing after the middle dimension).
Cairns had performed a number of computer calculations on some small finitely
generated nilpotent groups and had observed that their homology ranks obeyed
this one-mountain principle (he actually carried out equivalent calculations with
nilpotent lie algebras). Several people tried without success to answer this question
positively. Eventually Cairns and his colleagues [1] computed some significantly
larger examples and found a counter-example. They also obtained some new general estimates concerning the ranks of these homology groups.
This work of Cairns illustrates one of the ways where computer computations
can lead to intriguing questions, questions which are are hard to come by using
hand calculations alone. In this case, a negative solution was also found by machine
computations, leading to number of positive results.
8. Parafree groups
A group is termed parafree if it is residually nilpotent and has the same nilpotent factor groups as some fixed free group. Finitely presented, parafree groups
EXPERIMENTING AND COMPUTING WITH INFINITE GROUPS
11
which are not free are plentiful. The isomorphism problem is one of the many open
problems about these groups. Now G. Baumslag [2] has proved that the groups
Gi,j =< a, b, c; a = [ci , a][cj , b] > (i ≥ 1, j ≥ 1)
are all parafree. Moreover, if F denotes a free group of rank two, then
Gi,j /G00 ∼
= F/F 00 .
i,j
In 1990 a seminar was held a CCNY on computational group theory. The isomorphism problem was one of a number of concrete problems investigated during
the course of that seminar. One of the results of the seminar was that Lewis
and Liriano [9] undertook the computation of the cardinality of Hom(Gi,j , T ),
where T was chosen to be SL(2, Z/4Z) and SL(2, Z/5Z). These computations
yielded a number of interesting results. First, the number of elements in the sets
Hom(Gi,j , SL(2, Z/5Z)) were different for G1,1 and G2,2 . Consequently, G1,1 and
G2,2 are not isomorphic. In addition, they proved that there are more homomorphisms of G1,1 into SL(2, Z/4Z)) then there are of F into SL(2, Z/4Z)). This
implies that
∼
6 F/F 000 .
G1,1 /G000
1,1 =
Whether the Gi,j are distinguished by their subscripts is still unknown.
9. Some open problems
It seems possible that investigation by computer might shed some light on a
number of open problems. We describe a number of these here.
1. A random enumeration by Magnus produced a number of one-sixth groups
which did not seem to have confluent rewriting systems. This suggests the somewhat more general question: Does every hyperbolic group have a finite, confluent,
rewriting system?
2. A long standing unresolved question of Hanna Neumann is the following: let A
and B be two finitely generated subgroups of a free group F . Let α denote the rank
of A, β the rank of B and γ the rank of their intersection. Is
(γ − 1) ≤ α − a)(β − 1) ?
We have run a number of tests on this problem, some of which involved hundreds of
generators of length at least 100. All of the answers satisfied the above inequality.
So it seems reasonable to suggest that it is indeed valid.
3. We have already noted earlier that there exists a finitely presented group G
such that there is no algorithm which decides whether or not any finitely generated
subgroup of G is finitely related. However, this question is still open for one-relator
groups: is every finitely generated subgroup of a one-relator group finitely related?
Magnus contains the tools for enumerating the relations that hold between the
generators of a finitely generated subgroup of a finitely presented group. Gathering
such relations together in a number of special instances might provide the impetus
for making some progress towards solving this problem.
4. A problem related to Problem 3 is the following: if the finitely generated group G
contains a free normal subgroup with infinite cyclic quotient, is G finitely related?
5. There is an old problem about amalgamated products of free groups which is
probably due to Peter Neumann: if A and B are free and C is of finite index in
both, can A ?C B be a simple group?
12
GILBERT BAUMSLAG AND CHARLES F. MILLER III
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EXPERIMENTING AND COMPUTING WITH INFINITE GROUPS
13
Gilbert Baumslag, Department of Mathematics, City College of New York, Convent Ave, and 138th St., New York NY 10031, USA
E-mail address: gilbert@groups.sci.ccny.cuny.edu
Charles F. Miller III, Department of Mathematics, University of Melbourne, Parkville 3052, Australia
E-mail address: cfm@maths.mu.oz.au