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Abstract. The object of this paper is to describe a simple method for proving that

arXiv:math/9305201v1 [math.GR] 26 May 1993MUSINGS ON MAGNUSGilbert BaumslagAbstract. The object of this paper is to describe a simple method for proving thatcertain groups are residually torsion-free nilpotent, to describe some new parafreegroups and to raise some new problems in honour of the memory of Wilhelm Magnus.1.

IntroductionI first heard of Wilhelm Magnus in 1956, when I was attending some lecturesby B.H. Neumann on amalgamated products.

At some point during the courseof these lectures, Neumann remarked that Magnus was the first mathematician torecognize the value of amalgamated products and had shown just how effective atool they were, in his work on groups defined by a single relation. I was working,at that time, on an universal algebra variation of free groups, involving groupswith unique roots, which I called D-groups [1].

Consequently, in an attempt tofind analogues of various theorems about free groups for D-groups, I found myselfreading a beautiful paper of Magnus, in which he proved the residual torsion-freenilpotence of free groups. Much of my talk today will be concerned with these twotopics, one-relator groups and residual nilpotence.

Let me begin by reminding youof some of the definitions involved. Let P be a property of groups.Definition.

We say that a group G is residually a P-group if for each g ∈G, g ̸= 1,there exists a normal subgroup N of G, such that g /∈N and G/N has P.The properties that I will be mainly concerned with here are freeness, nilpotenceand torsion-free nilpotence. I will make use of the usual commutator notation.

Thusif H and K are subgroups of a group G then[H, K] = gp(h−1k−1hk | h ∈H, k ∈K)is the subgroup generated by all the commutators h−1k−1hk. The lower centralseries of G is defined to be the seriesG = γ1(G) ≥γ2(G) ≥· · · ≥γn(G) ≥.

. .

,where γn+1(G) = [γn(G), G]. G is termed nilpotent if γc+1(G) = 1 for some c. I amnow in a position to formulate the theorem of Magnus [12] that I alluded to before.1991 Mathematics Subject Classification.

Primary 20F05, 20F14.Key words and phrases. Residually torsion-free nilpotent groups, one-relator groups, D-groups.The author was supported in part by NSF Grant #9103098Typeset by AMS-TEX1

2GILBERT BAUMSLAGTheorem 1. Free groups are residually torsion-free nilpotent.The basic idea involved in the proof of Theorem 1 is beautifully simple.

Magnusconcocts a faithful representation of a given free group F in the group of units of acarefully chosen ring R with 1. Each of the elements f ∈F takes the form 1 + φ,where φ lies in an ideal R+ of R. R+ carries with it the structure of a metric spacedesigned so as to ensure that if f ∈γn(F), then d(φ, 0) ≤2−n, where here d(φ, 0)denotes the distance between φ and 0.

This suffices to ensure that∞\n=1γn(F) = 1.The sketch of the proof, below, amplifies these remarks.Proof. Let F be free on x1, .

. .

, xq. Consider the ring R of power series in the non-commuting indeterminates ξ1, .

. .

, ξq with rational coefficients. Each element r ∈Rcan be thought of as an infinite sum which takes the formr = r0 + r1 + · · · + rn + .

. .

,where r0 ∈Q and the homogeneous component rn of r of degree n > 0 is a finitesum of rational multiples cξi1 . .

. ξin of monomials ξi1 .

. .

ξin of degree n, c ∈Q, ij ∈{1, . .

. , q}.

Here ξi1 . .

. ξin = ξj1 .

. .

ξjm only if n = m and ir = jr for each r =1, . .

. , n. Each of the elements ai = 1 + ξi is invertible in R, with inverse a−1i=1 −ξi + ξ2i −ξ3i + .

. .

. It then turns out that the elements a1, .

. .

, aq freely generatea free subgroup of rank n of the multiplicative group of units of R. We define R+to be the ideal of R consisting of those elements r ∈R such that r0 = 0. We definea metric d on R+ by setting d(r, 0) = 0 if r = 0 and, if r ̸= 0,d(r, 0) = 2−nwhere n is the degree of the first non-zero homogeneous component of r. Magnusproves that if f = 1 + φ ∈γn(F), then all of the homogeneous components f1 =f2 = · · · = fn−1 = 0, i.e., d(φ, 0) ≤2−n.

It is not hard then to deduce that F isresidually torsion-free nilpotent.One of the many consequences of this theorem of Magnus is the following charac-terisation of finitely generated free groups, which Magnus proved a few years laterin [13]Theorem 2. If the group G can be be generated by q elements, where q is finite,and if G/γn(G) ∼= F/γn(F), where F is a free group of rank q, then G ∼= F.Theorem 2 will play a role here in due course.2.

One-relator groupsMy next encounter with Magnus came some years later. After spending a post-doctoral year in Manchester, as a Special Lecturer, I came to Princeton in 1959as an instructor.

Some time towards the end of that year, Trueman MacHenry, adoctoral student of Magnus whom I had met in Manchester, came to visit me inPrinceton. He was accompanied by Bruce Chandler, another of Magnus’ doctoral

MUSINGS ON MAGNUS3students. MacHenry brought greetings from Magnus and an implicit offer of a jobat the Courant Institute.

I was very happy at the prospect of working with Magnus,and came up to New York to give a talk early in 1960. This was the first time that Ihad actually met Magnus and I was very impressed with his quickness and his vastknowledge, not only of mathematics but of almost everything else as well.

He musthave been amused by my talk, which was to an audience which consisted mainlyof analysts, because I talked about an extremely esoteric theorem that NormanBlackburn and I [4] had proved about 9 months earlier, a theorem that only themost special of specialists would have found interesting. Nonetheless, the analystswere apparently convinced by Magnus that I should be offered a position and theoffer was made explicit soon after.

I immediately accepted the position and came tothe Courant in the late summer of 1960. I spent part of that summer in Pasadena,where I met Danny Gorenstein and Roger Lyndon.

Roger Lyndon had been work-ing on groups with parametric exponents [9]. There were other things on his mindas well and he asked me a number of questions about one-relator groups while wewere in Pasadena.

I had not read Magnus’s papers on one-relator groups, and soI was forced to spend part of that summer trying to understand Magnus’s work.There are two main theorems that I would like to describe. The first of these isMagnus’ celebrated ”Freiheitssatz” [10].Theorem 3.

LetG =< x1, . .

. , xq; r = 1 >be a group defined by a single relator r. If the first and last letters of r are notinverses and if x1 appears in r, then the subgroup of G generated by x2, .

. .

, xq is afree group, freely generated by x2, . .

. , xq.One of the byproducts of the proof of the Freheitssatz was an extraordinaryunravelling of the structure of these groups, which allowed Magnus to deduce, indue course, that one-relator groups have solvable word problem [11]:Theorem 4.

Let G be a group defined by a single relation. Then G has a solvableword problem.3.

Surface groupsAs I remarked earlier, I came to the Courant Institute in the late summer of 1960.Magnus was part of the electromagnetic group of Morris Kline and used to go toa seminar on electromagnetism. I remember one occasion, after tea, when MorrisKline and Magnus and I were in the elevator, in the old hat factory which hadbecome the Courant Institute.

Kline was talking to Magnus and addressed him asBill. I saw Magnus shudder at the prospect of being called Bill.

Magnus was muchtoo polite to say anything, but I did not feel at all constrained to be silent. So I saidsomething like this to Kline: ”Morris, Wilhelm is a Geheimrat and so he simplycannot be called Bill.Wilhelm is more appropriate”.

Morris Kline smiled andindicated that this was okay with him. To which Magnus responded by mutteringunder his breath: ”Thank God”.

He later told me that he had hated being addressedas Bill, and was grateful to me for telling Kline to call him Wilhelm.Magnushad a large group of Ph.D. students in 1961. They included Karen Fredericks,Bruce Chandler, Seymour Lipschutz and Trueman MacHenry, all of whom workedwith Magnus on Combinatorial Group Theory.

Martin Greendlinger had already

4GILBERT BAUMSLAGcompleted his beautiful work on small cancellation theory, but was still around. Inaddition, Magnus had some other students who worked with him on Hill’s equation.There was always a line of students outside his office and I offered to take onsome of them to relieve him of some of the burden.Magnus ran a seminar onCombinatorial Group Theory at the time.The seminar was a mixture of pureresearch and reports by students on papers that they were reading.

Shortly afterI arrived, the research part of the seminar was broadened and the role of studentswas reduced essentially to zero. One of Magnus’ habits was to propose a numberof problems in the seminar.

He was always interested in purely algebraic proofsof theorems that had been proved by other means. In particular, he asked late in1961, whether there was a direct proof of the residual finiteness of the fundamentalgroups of two-dimensional orientable surfaces.

Both Karen Frederick and I beganto work on this problem and we both eventually, independently, came up with asolution. Her solution [7] was very closely tied to the actual presentation of thesesurface groups.

I took a somewhat more general approach which yielded somewhatmore [2]. The net result was the followingTheorem 5.

Let F be a free group on X, A a free abelian group on Y, f an elementof F which is not a proper power in F, and a an element of A which is not a properpower in A. Then the groupG =< X ∪Y ; [y, y′] = 1(y, y′ ∈Y ), f = a >is residually free.In particular, it follows from this theorem that surface groups are residually freeand hence residually torsion-free nilpotent (and also residually finite).

The residualnilpotence of one-relator groups was itself a topic of some interest to Magnus. Helater put Bruce Chandler to work on this topic.

Chandler [6] eventually found analternative proof of the residual torsion-freeness of surface groups by making useof the ring R that I described at the outset. Some years later I found yet anothermeans of proving the residual nilpotence of surface groups.

I never discussed thismethod with Magnus, and subsequently forgot about it.A few months ago, J.Lewin told me that he had also found a very simple argument to prove the residualnilpotence of surface groups. This prompted me to rethink some of my old ideasand it is to these thoughts that I want to turn next.4.

Residually nilpotent groupsThe fundamental groups of two dimensional orientable surfaces all contain a freesubgroup with infinite cyclic factor group. Thus they have the same form as thegroups covered by the following theorem.Theorem 6.

Let G be a finitely generated group.Suppose that G contains afree, normal subgroup N such that G/N is infinite cyclic. If G/γ2(N) is residu-ally torsion-free nilpotent, then so is G.This theorem is similar in spirit to a theorem of P. Hall [8], who proved that if Gis a group with a normal, nilpotent subgroup N, then G is nilpotent if G/γ2(N) isnilpotent.

Theorem 6 leads to a host of new examples of residually nilpotent groups.In particular, it can be used to give yet another proof of the residual torsion-freenilpotence of surface groups. There is a further use of Theorem 6 that I want todescribe here.

To this end, let me recall the following definition.

MUSINGS ON MAGNUS5Definition. A group G is termed parafree if G is residually nilpotent and thereexists a free group F such thatG/γn(G) ∼= F/γn(F) for all n.Parafree groups, which can be likened to free groups, exist in profusion, see,e.g., [3].

It should be pointed out that it follows from Magnus’ Theorem 2 that anon-free, finitely generated parafree group G with the same nilpotent factor groupsG/γn(G) as a free group of rank q cannot be generated by q elements. It is notknown how closely a parafree group can resemble a free group.

I want to describenext some new, non-free parafree groups, which very closely resemble free groups.The proof that these groups are parafree is an easy application of Theorem 5 (see6).Theorem 7. Let F be the free group on s, t, a1, .

. .

, aq and let w be an element ofF which involves a1 and does not involve s. In addition suppose that w lies in thek −th term F (k) of the derived series of F. Then the one-relator groupGw =< s, t, a1, . .

. , aq; a1 = ws−1t−1st >is parafree and not free.

MoreoverG/G(k)w∼= H/H(k),where H is a free group of rank q + 1.I will sketch the proofs of Theorems 6 and 7 in section 6.5. Some problems on D-groupsLet p1 = 2, p2 = 3, .

. .

be the set of all primes in ascending order of magnitude.Definition. A group G is called a D-group if it admits a set of unary operatorsπ1, π2, .

. .such that for all g ∈Ggpiπi = g = (gπi)πi.It is not hard to verify that these D-groups consist precisely of those groups G inwhich extraction of n-th roots is uniquely possible, for every positive integer n. Theclass of all D-groups form a variety of (universal) algebras.

The precise technicaldescription of these terms does not matter. It suffices only to say that in such avariety one has the notion of a free D-group, as well as all of the other notions thatone makes use of in group theory.

There are a number of properties of such freeD-groups that are similar to properties of free groups. For example one has thefollowing

6GILBERT BAUMSLAGTheorem 8. D-subgroups of free D-groups are free.I was told about this theorem by Tekla Taylor-Lewin, but I have not been ableto locate a reference.

Magnus was fond of these D-groups, and discussed them inhis book with Karrass and Solitar [14]. It seems appropriate, therefore, to raisehere some new problems about free D-groups, in his memory.

To this end, let Fbe the free D-group onx1, . .

. , xq.It is not hard to see that the mappingxi 7→1 + ξi (i = 1, .

. .

, q)defines a homomorphism φ of F into the group of units of R.Problem 1. Is φ a monomorphism?Problem 2.

LetG =< x1, . .

. , xq; r = 1 > .If extraction of n-th roots in G is unique, whenever such roots exist, can G beembedded in a D-group?Problem 3.

Suppose that the one-relator groupG =< x1, . .

. , xq; r = 1 >can be embedded in a D-group.

If H is the one-relator D-group generated, as aD-group, by x1, . .

. , xq and defined, as a D-group by the single relation r = 1, is theword problem solvable for H?

In general, is the word problem solvable for one-relatorD-groups?Problem 4. Is there a freiheitssatz for one-relator D-groups?Problem 5.

Can free groups be characterised by a length function?Problem 6. Does a free D-group act freely on a Λ-tree, for a suitable choice of Λ?6.

ProofsI want to sketch here the proofs of Theorem 6 and Theorem 7. I would like tobegin with the proof of Theorem 6.

I will adopt the notation used in the statementof the theorem. Since G/N is infinite cyclic, we can choose an element t ∈G, suchthat G = gp(N, t).

So t is of infinite order modulo N. Suppose that g ∈G, g ̸= 1.We want to find a normal subgroup K of G such that G/K is torsion-free nilpotentand g /∈K. It is clear that it suffices to consider the case where g ∈N.

Since Nis free there exists an integer k such that g /∈γk(G). Notice that H = N/γk(N) isa torsion-free nilpotent group and so we can form the Malcev completion H of H(see [15]).

This group H is a minimal D-group containing H. It is again nilpotentof class k −1 and torsion-free. Moreover if we denote by τ the automorphism thatt induces on H, then τ extends uniquely to an automorphism τ of H. Let G bethe semidirect product of H with the infinite cyclic group generated by an elementt, where t acts on H as τ.

Then M = H/γ2(H) is a direct sum of copies of theadditive group of Q (see [1]). Morevover, it is not hard to deduce from the fact

MUSINGS ON MAGNUS7that G/γ2(N) is residually torsion-free nilpotent, that G/γ2(H) is also residuallytorsion-free nilpotent. We now view M as a module over the rational group algebraΓ of the infinite cyclic group on t. Since Γ is a principal ideal domain and the groupG is finitely generated, the Γ-module M is finitely generated and hence a directsum of cyclic modules.

This decomposition of M makes it possible to understandthe action of t on H and, using the fact that H = F/γk(F), to deduce that His residually torsion-free nilpotent. It is easy to deduce that G is itself residuallytorsion-free nilpotent.

There are three steps in the proof of Theorem 7. The firstmakes use of Magnus’ method of unravelling the structure of one-relator groups.This method allows to prove, easily, that the normal closure N of s, a1, .

. .

, aq inGw is free. The second step involves the verification that G/γ2(N) is residuallytorsion-free nilpotent.

So, by Theorem 6, Gw is residually torsion-free nilpotent.The other properties of Gw follow directly from the form of the defining relationof Gw. The final step in the proof of Theorem 7 is the verification that Gw is notfree.

This is accomplished by invoking an algorithm of J.H.C. Whitehead [15].

Ihave only talked here about a few of Magnus’ theorems. All of his work is filledwith beautiful, new ideas, giving joy to all of us.

Wilhelm Magnus is sorely missed,but his work will be with us always.References1. Gilbert Baumslag, Some aspects of groups with unique roots, Acta Mathematica 104 (1960),217–303.2., On generalised free products, Math.

Zeitschrift 78 (1962), 423–438.3., More groups that are just about free, Bull. Amer.

Math. Soc.

72 (1968), 752–754.4. Gilbert Baumslag and Norman Blackburn, Groups with cyclic upper central factors, Proc.London Math.

Soc. (3) 10 (1960), 531–544.5.

Gilbert Baumslag and Bruce Chandler (eds. ), Wilhelm Magnus Collected Papers, SpringerVerlag, New York, Heidelberg, Berlin, 1984.6.

B. Chandler, The representation of a generalized free product in an associative ring, Comm.Pure Appl. Math.

21 (1968), 271–288.7. K.N.

Frederick, The Hopfian property for a class of fundamental groups, Comm. Pure Appl.Math.

16 (1963), 1–8.8. P. Hall, Some sufficient conditions for a group to be nilpotent, Illinois J.

Math. 2 (1958),787–801.9.

R.C. Lyndon, Groups with parametric exponents, Trans.

American Math. Soc.

96 (1960),445–457.10. W. Magnus, ¨Uber diskontinuierliche Gruppen mit einer definierenden Relation (Der Frei-heitssatz), J. Reine Angew.

Math. 163 (1930), 141–165.11., Das Identit¨atsproblem f¨ur Gruppen mit einer definierenden Relation, Math.

Ann.106 (1932), 295–307.12., Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann.111 (1935), 259–280.13., ¨Uber freie Faktorgruppen und freie Untergruppen gegebener Gruppen, Monatsh.

Math.47 (1939), 307–313.14. W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Wiley, New York, 1966.15.

A.I. Mal’cev, On a class of homogeneous spaces, Izvestiya Akad.

Nauk SSSR Ser. Mat.

13(1949), 9–32; English transl. in Amer.

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Transl. (1) 9 (1962), 276–307.16.

J.H.C. Whitehead, On equivalent sets of elements in a free group, Ann.

of Math. 37 (1936),782–800.Department of Mathematics, City College of New York, New York, N.Y. 10031

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