Some aspects of one-relator groups

Newman, B.B. (1968) Some aspects of one-relator groups. PhD thesis, University College of Townsville.

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Abstract

For an introduction to the theory of one-relator groups se Baumslag 1964, and Magnus, Karrass, and Solitar 1966. There are three main themes in the work which follows.

The first theme is the determination of the Abelian subgroups of a one-relator group. This investigation was prompted by a conjecture of Baumslag 1964 that the additive group of rationals is not a subgroup of a one-relator group. The Abelian subgroups of one-relator groups have now been completely determined; they are free Abelian of rank ≤ 2 or the additive group of nadic rationals, n a positive integer or finite cyclic groups. Theorem: (See Theorem 1.2.3) Let G = gp(a, b, c, … | R) be a torsion-free one-relator group. Then no non-trivial element has more than finitely many prime divisors. Moreover a non-trivial element is not divisible by more than finitely many powers of a prime p, if p is greater than the length of the relator.

Thus the additive group of rationals is not a subgroup of a torsion-free one-relator group, in fact; Corollary: (See Corollary 1.2.4) The additive group of rationals is not a subgroup of a one-relator group.

In the case of one-relator groups with torsion, one can say much more. Theorem: (See Theorem 2.3.2) The Abelian subgroups of a one-relator group with torsion are cyclic. Corollary: (See Corollary 2.3.3) The soluble subgroups of a one-relator group with torsion are cyclic. Corollary: (See Corollary 2.3.4) The centralizer of every non-trivial element of a one-relator group with torsion is cyclic.

The second theme is the problem of extending the Freiheitssatz. This theorem, proved by Magnus 1930, is the basic result in the theory of one-relator groups. Let

G = gp(a, b, c, … | R(a, b, c, …))

where R is cyclically reduced, and suppose the generator a occurs non-trivially in R. Then the Freiheitssatz states that b, c, ... freely generate a subgroup of G. But something more than this is true, and we seek to extend the Freiheitssatz by proving that, for some integer m

a^m, b, c, …

freely generate a subgroup of G in certain rather general cases. The first result in this direction was obtained for certain two generator groups by Mendelsohn and Ree 1967. Here we prove the following. Theorem: (See Theorem 1.3.11) Let G = gp(a, b, c, … | R) where a, b, c occur non-trivially in R with σa(R) = 0. Then for one of the generators a, b, or c, (say b) there exists an integer m such that, for all integers α > m

a, b^α, c, …

freely generate a subgroup of G.

Again one can say more for groups with torsion. The basic result for such groups is the Spelling Theorem. (See Theorem 2.1,1) Let G = gp(a, b, … | R^n) n > 1, where R is cyclically reduced. Suppose that two words W(a, b, …), V(b, …), where W is a freely reduced word containing a non-trivially and V does not contain a, define the same element of G. Then W contains a subword which is identical with a subword of R±^n of length greater than (n - 1)/n times the length of R^n.

From this theorem one can prove the following extension of the Freiheitssatz. Corollary: (See Corollary 2.1,6) Let G = gp(a, b, c, … |R^n) n > 1, where R is cyclically reduced involving a, b non-trivially, and suppose β is any integer which does not divide the a-exponents in R^n. Then a^β, b, c, … freely generate a subgroup of G.

The third theme is concerned with algorithmic problems in one-relator groups; more specifically we are primarily concerned with the word problem and the conjugacy problem in one-relator groups with torsion. These algorithmic problems, proposed by Dehn 1911 are fundamental problems in the presentation theory of groups. In general they are unsolvable. For free groups both the word problem and the conjugacy problem are solvable. In 1932 Magnus used an ingenious application of the Freiheitssatz to prove that one-relator groups have a solvable word. problem. However Magnus, Karrass, and Solitar 1966 pointed out that there are some unsatisfactory aspects of the solution, for the algorithm appears to be unnecessarily complicated. In the case of less than one-sixth groups investigated by Greendlinger 1960 there is a simpler algorithm. Using the Spelling Theorem a trivial proof of the solvability of the word problem for one-relator groups with torsion can be given, (See Corollary 2.14) and the algorithm which emerges is of the required degree of simplicity, and provides a bridge between the work of Magnus and that of Tartakovskii 1949 and Greendlinger 1960. Another similar problem related to a problem of Lyndon 1962 is the following: Corollary: (See Corollary 2.1.5) Let G = gp(a, b, c, … | R^n) n > 1 and let W, Z be subsets of the generators. Then there is an algorithm to determine for an arbitrary element g ε G if g = w(W)z(Z) for some words w, z.

As for the conjugacy problem very little has been done. Greendlinger 1960b, 1964 has proved the solvability of the conjugacy problem for less than one-sixth groups, and Soldatova 1967 has extended the result to certain less than one-fourth groups. The conjugacy problem for the free product of two free groups with a cyclic subgroup amalgeated has been solved by Lipschutz (unpublished). In this work we show that all one-relator groups with torsion have a solvable conjugacy problem. Theorem: (See Theorem 3.2,3) Let G be a one-relator group with torsion. Then the conjugacy problem and the extended conjugacy problem relative to the subgroup generated by any subset of the generators are solvable in G.

The problem of finding an algorithm to determine whether or not an arbitrary element of a group is a power has been investigated by Reinhart 1962 and by Lipschutz 1965 and 1968. We prove the following result. Theorem: (See Theorem 3.3.1) Let G = gp(a, b, c, ... | R^n) n > 1. Given g ε G there is an algorithm to determine the roots of g.

There are a few miscellaneous results which emerge in the work. The first concerns the Frattini subgroup. Theorem: (See Theorem 1 .3.12) A one-relator group has trivial Frattini subgroup if (a) it is torsion-free with more than two generators or (b) it has torsion with more than one generator. The next result concerns a residual property, Corollary: (See Corollary 2.1.7) Let G = gp(a, b, c, … | R^n) n > 1. Then G is residually a two-generator one-relator group with torsion. The following theorem of Baumslag' and. Steinberg 1964 may be proved quite easily. Theorem: (See Theorem 1.3.7) Let w(x1 , x2, …, Xn) be an element of a free group F freely generated by x1, x2, …, xn, x which is neither a proper power nor a primitive. If g1, g2, …, gn, g, generate a free group G and are connected by the relation w(g1, g2, …, gn) = g^m m > 1 then the rank of G is at most n - 1.

There is a common strata to these three main themes, namely the technique for proving them. The technique is as follows. In a one-relator group with more than one generator occurring non-trivially in the cyclically reduced relator one can, without too much disruption of the group, arrange for the exponent sum on one of the generators to be zero. Let G be such a one-relator group, and let N be the normal subgroup of G generated by the remaining generators. This is usually a complicated group, infinitely generated and infinitely related. But it has one nice property, it is the direct limit of a chain of subgroups of N, N0, gp(N0,N1), gp(N-1 ,N0, N1), gp(N-1,N0,N1,N2), … where the Ni are isomorphic one-relator groups, with the length of the relator less than that of the original group G. We thus have a basis for an induction argument to prove the one-relator group G has some specified group theoritic property P. Thus the induction hypothesis would be that all one-relator groups with relator length less than the length of the relator of G have the property P. Now the normal subgroup N is well situated in G, for it has infinite cyclic factor group. For the properties of interest here, we can show that in order to prove G has the property P it will suffice to prove that N has the property P. We now use the nice structure of N. By the induction hypothesis N0, and in fact each Ni has the property P. Thus the first term in the chain above has property P.

Does the second term, gp(N0, N1) have property P? Well the gp(N0, N1) is a generalized free product of N0 and N1 amalgamating a subgroup generated by a common subset of the generators of N0 and N1, so the basic problem is this; when does a generalized free product of two groups each having the property P, have property P. For example it is known (Neumann 1954) that the generalized free product of torsion-free groups is torsion-free. In order for some property P of the factors to be inherited by a free product with amalgamation it is usually necessary to put conditions on the amalgamated subgroup. For example the generalized free product of residually finite groups is residually finite if the amalgamated subgroup is finite. Thus we will have to find for each property P (and there will be a different one in each chapter) an appropriate type of subgroup, call it a q-subgroup, such that the following proposition holds. Proposition 1: If C = {A * B ; J} is the generalized free product of the factors A and B amalgamating the subgroup J, and A and B have the property P, and J is a q-subgroup of A and B, then C has the property P.

For the groups with which we are concerned we know from the Freiheitssatz that the amalgamated subgroup is free. But freeness is not usually sufficient. In our case we will need the following proposition: Proposition 2: Any subset of the generators of a one-relator group G generates a q-subgroup of G. The Propositions 1 and 2 will then allow us to take one step up the chain and prove gp(N0, N1) has the property P.

Now the third term of the chain, gp(N-1, N0, N1) is again a generalized free product, of the two factors gp(N0, N1) and N -1 amalgamating a subgroup J-1 generated by a common subset of the generators of N0 and N-1. From the two preceding paragraphs we know that both these factors have the property P. All we need in view of Proposition 1 is for the amalgamated subgroup to be a q-subgroup of both factors. From Proposition 2, J-1 is a q-subgroup of N-1 . In order for the amalgamated subgroup to be a q-subgroup of the first factor it suffices to have the following result. Proposition 3: A q-subgroup of a q-subgroup is a q-subgroup, and if C = {A * B ; J} where the amalgamated subgroup J is a q-subgroup of the factors A and B, then the factors A and B are q-subgroups of C. With this result we can proceed to the third term in the chain. For N0 is by Proposition 3 a q-subgroup of gp(N0, N1) and J-1 is by Proposition 2 a q-subgroup of N0, hence J-1 is a q-subgroup of gp(N0, N1). One continues stepping up the chain by repeating these arguments, for each successive term is formed by a similar generalized free product construction. Hopefully the direct limit of the chain of groups with property P will also have property P. This then would prove that N has the property P.

Item ID: 58078
Item Type: Thesis (PhD)
Keywords: Newman's Spelling Theorem, one-relator groups, group theory
Copyright Information: Copyright 1968 B.B. Newman.
Additional Information:

"A dissertation submitted to the University of Queensland in partial fulfillment of the requirements for the degree of Doctor of Philosophy"--Title page. In 1968 University of Queensland encompassed University College of Townsville, now known as James Cook University.

Date Deposited: 18 Apr 2019 01:28
FoR Codes: 01 MATHEMATICAL SCIENCES > 0101 Pure Mathematics > 010105 Group Theory and Generalisations @ 100%
SEO Codes: 97 EXPANDING KNOWLEDGE > 970101 Expanding Knowledge in the Mathematical Sciences @ 100%
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