In the mathematical subject of group theory, a one-relator group is a group given by a group presentation with a single defining relation. One-relator groups play an important role in geometric group theory by providing many explicit examples of finitely presented groups.

Formal definition

A one-relator group is a group G that admits a group presentation of the form

 

 

 

 

(1)

where X is a set (in general possibly infinite), and where is a freely and cyclically reduced word.

If Y is the set of all letters that appear in r and then

For that reason X in (1) is usually assumed to be finite where one-relator groups are discussed, in which case (1) can be rewritten more explicitly as

 

 

 

 

(2)

where for some integer

Freiheitssatz

Let G be a one-relator group given by presentation (1) above. Recall that r is a freely and cyclically reduced word in F(X). Let be a letter such that or appears in r. Let . The subgroup is called a Magnus subgroup of G.

A famous 1930 theorem of Wilhelm Magnus,[1] known as Freiheitssatz, states that in this situation H is freely generated by , that is, . See also[2][3] for other proofs.

Properties of one-relator groups

Here we assume that a one-relator group G is given by presentation (2) with a finite generating set and a nontrivial freely and cyclically reduced defining relation .

  • A one-relator group G is torsion-free if and only if is not a proper power.
  • A one-relator presentation is diagrammatically aspherical.[5]
  • If is not a proper power then a one-relator group G has cohomological dimension .
  • A one-relator group G is free if and only if is a primitive element; in this case G is free of rank n  1.[7]
  • Suppose the element is of minimal length under the action of , and suppose that for every either or occurs in r. Then the group G is freely indecomposable.[8]
  • If is not a proper power then a one-relator group G is locally indicable, that is, every nontrivial finitely generated subgroup of G admits a group homomorphism onto .[9]
  • If G is a one-relator group and is a Magnus subgroup then the subgroup membership problem for H in G is decidable.[10]
  • A one-relator group G given by presentation (2) has rank n (that is, it cannot be generated by fewer than n elements) unless is a primitive element.[11]
  • Let G be a one-relator group given by presentation (2). If then the center of G is trivial, . If and G is non-abelian with non-trivial center, then the center of G is infinite cyclic.[12]
  • Let where . Let and be the normal closures of r and s in F(X) accordingly. Then if and only if is conjugate to or in F(X).[13][14]
  • There exists a finitely generated one-relator group that is not Hopfian and therefore not residually finite, for example the Baumslag–Solitar group .[15]
  • Let G be a one-relator group given by presentation (2). Then G satisfies the following version of the Tits alternative. If G is torsion-free then every subgroup of G either contains a free group of rank 2 or is solvable. If G has nontrivial torsion, then every subgroup of G either contains a free group of rank 2, or is cyclic, or is infinite dihedral.[16]
  • Let G be a one-relator group given by presentation (2). Then the normal subgroup admits a free basis of the form for some family of elements .[17]

One-relator groups with torsion

Suppose a one-relator group G given by presentation (2) where where and where is not a proper power (and thus s is also freely and cyclically reduced). Then the following hold:

  • The element s has order m in G, and every element of finite order in G is conjugate to a power of s.[18]
  • Every finite subgroup of G is conjugate to a subgroup of in G. Moreover, the subgroup of G generated by all torsion elements is a free product of a family of conjugates of in G.[4]
  • G admits a torsion-free normal subgroup of finite index.[4]
  • Newman's "spelling theorem"[19][20] Let be a freely reduced word such that in G. Then w contains a subword v such that v is also a subword of or of length . Since that means that and presentation (2) of G is a Dehn presentation.
  • G has virtual cohomological dimension .[21]
  • G is coherent, that is every finitely generated subgroup of G is finitely presentable.[23]
  • The isomorphism problem is decidable for finitely generated one-relator groups with torsion, by virtue of their hyperbolicity.[24]

Magnus–Moldavansky method

Starting with the work of Magnus in the 1930s, most general results about one-relator groups are proved by induction on the length |r| of the defining relator r. The presentation below follows Section 6 of Chapter II of Lyndon and Schupp[26] and Section 4.4 of Magnus, Karrass and Solitar[27] for Magnus' original approach and Section 5 of Chapter IV of Lyndon and Schupp[28] for the Moldavansky's HNN-extension version of that approach.[29]

Let G be a one-relator group given by presentation (1) with a finite generating set X. Assume also that every generator from X actually occurs in r.

One can usually assume that (since otherwise G is cyclic and whatever statement is being proved about G is usually obvious).

The main case to consider when some generator, say t, from X occurs in r with exponent sum 0 on t. Say in this case. For every generator one denotes where . Then r can be rewritten as a word in these new generators with .

For example, if then .

Let be the alphabet consisting of the portion of given by all with where are the minimum and the maximum subscripts with which occurs in .

Magnus observed that the subgroup is itself a one-relator group with the one-relator presentation . Note that since , one can usually apply the inductive hypothesis to when proving a particular statement about G.

Moreover, if for then is also a one-relator group, where is obtained from by shifting all subscripts by . Then the normal closure of in G is

Magnus' original approach exploited the fact that N is actually an iterated amalgamated product of the groups , amalgamated along suitably chosen Magnus free subgroups. His proof of Freiheitssatz and of the solution of the word problem for one-relator groups was based on this approach.

Later Moldavansky simplified the framework and noted that in this case G itself is an HNN-extension of L with associated subgroups being Magnus free subgroups of L.

If for every generator from its minimum and maximum subscripts in are equal then and the inductive step is usually easy to handle in this case.

Suppose then that some generator from occurs in with at least two distinct subscripts. We put to be the set of all generators from with non-maximal subscripts and we put to be the set of all generators from with non-maximal subscripts. (Hence every generator from and from occurs in with a non-unique subscript.) Then and are free Magnus subgroups of L and . Moldavansky observed that in this situation

is an HNN-extension of L. This fact often allows proving something about G using the inductive hypothesis about the one-relator group L via the use of normal form methods and structural algebraic properties for the HNN-extension G.

The general case, both in Magnus' original setting and in Moldavansky's simplification of it, requires treating the situation where no generator from X occurs with exponent sum 0 in r. Suppose that distinct letters occur in r with nonzero exponents accordingly. Consider a homomorphism given by and fixing the other generators from X. Then for the exponent sum on y is equal to 0. The map f induces a group homomorphism that turns out to be an embedding. The one-relator group G' can then be treated using Moldavansky's approach. When splits as an HNN-extension of a one-relator group L, the defining relator of L still turns out to be shorter than r, allowing for inductive arguments to proceed. Magnus' original approach used a similar version of an embedding trick for dealing with this case.

Two-generator one-relator groups

It turns out that many two-generator one-relator groups split as semidirect products . This fact was observed by Ken Brown when analyzing the BNS-invariant of one-relator groups using the Magnus-Moldavansky method.

Namely, let G be a one-relator group given by presentation (2) with and let be an epimorphism. One can then change a free basis of to a basis such that and rewrite the presentation of G in this generators as

where is a freely and cyclically reduced word.

Since , the exponent sum on t in r is equal to 0. Again putting , we can rewrite r as a word in Let be the minimum and the maximum subscripts of the generators occurring in . Brown showed[30] that is finitely generated if and only if and both and occur exactly once in , and moreover, in that case the group is free. Therefore if is an epimorphism with a finitely generated kernel, then G splits as where is a finite rank free group.

Later Dunfield and Thurston proved[31] that if a one-relator two-generator group is chosen "at random" (that is, a cyclically reduced word r of length n in is chosen uniformly at random) then the probability that a homomorphism from G onto with a finitely generated kernel exists satisfies

for all sufficiently large n. Moreover, their experimental data indicates that the limiting value for is close to .

Examples of one-relator groups

  • Free abelian group
  • Baumslag–Solitar group where .
  • Torus knot group where are coprime integers.
  • Baumslag–Gersten group
  • Oriented surface group where and where .
  • Non-oriented surface group , where .

Generalizations and open problems

  • If A and B are two groups, and is an element in their free product, one can consider a one-relator product .
  • The so-called Kervaire conjecture, also known as Kervaire–Laudenbach conjecture, asks if it is true that if A is a nontrivial group and is infinite cyclic then for every the one-relator product is nontrivial.[32]
  • Klyachko proved the Kervaire conjecture for the case where A is torsion-free.[33]
  • A conjecture attributed to Gersten[22] says that a finitely generated one-relator group is word-hyperbolic if and only if it contains no Baumslag–Solitar subgroups.
  • If G is a finitely generated one-relator group (with or without torsion), is a torsion-free subgroup of finite index and is an epimorphism then has cohomological dimension 1 and therefore, by a result of Stallings, is locally free.[34] Baumslag, with co-authors, showed that in many cases, by a suitable choice of H and one can prove that that is actually free (of infinite rank).[35][36] These results led to a conjecture[22] that every finitely generated one-relator group with torsion is virtually free-by-cyclic.

See also

Sources

  • Wilhelm Magnus, Abraham Karrass, Donald Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations, Reprint of the 1976 second edition, Dover Publications, Inc., Mineola, NY, 2004. ISBN 0-486-43830-9. MR2109550

References

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  2. Lyndon, Roger C. (1972). "On the Freiheitssatz". Journal of the London Mathematical Society. Second Series. 5: 95–101. doi:10.1112/jlms/s2-5.1.95. hdl:2027.42/135658. MR 0294465.
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