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

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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

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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

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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]
  • 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

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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]
  •   is virtually free-by-cyclic, i.e.   has a subgroup   of finite-index such that there is a free normal subgroup   with cyclic quotient  .[26]

Magnus–Moldavansky method

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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[27] and Section 4.4 of Magnus, Karrass and Solitar[28] for Magnus' original approach and Section 5 of Chapter IV of Lyndon and Schupp[29] for the Moldavansky's HNN-extension version of that approach.[30]

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

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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[31] 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[32] 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

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  • 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

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  • 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.[33]
  • Klyachko proved the Kervaire conjecture for the case where A is torsion-free.[34]
  • 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.

See also

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Sources

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  • 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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  1. ^ Magnus, Wilhelm (1930). "Über diskontinuierliche Gruppen mit einer definierenden Relation. (Der Freiheitssatz)". Journal für die reine und angewandte Mathematik. 1930 (163): 141–165. doi:10.1515/crll.1930.163.141. MR 1581238. S2CID 117245586.
  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.
  3. ^ Weinbaum, C. M. (1972). "On relators and diagrams for groups with one defining relation". Illinois Journal of Mathematics. 16 (2): 308–322. doi:10.1215/ijm/1256052287. MR 0297849.
  4. ^ a b c Fischer, J.; Karrass, A.; Solitar, D. (1972). "On one-relator groups having elements of finite order". Proceedings of the American Mathematical Society. 33 (2): 297–301. doi:10.2307/2038048. JSTOR 2038048. MR 0311780.
  5. ^ Lyndon & Schupp, Ch. III, Section 11, Proposition 11.1, p. 161
  6. ^ Dyer, Eldon; Vasquez, A. T. (1973). "Some small aspherical spaces". Journal of the Australian Mathematical Society. 16 (3): 332–352. doi:10.1017/S1446788700015147. MR 0341476.
  7. ^ Magnus, Karrass and Solitar, Theorem N3, p. 167
  8. ^ Shenitzer, Abe (1955). "Decomposition of a group with a single defining relation into a free product". Proceedings of the American Mathematical Society. 6 (2): 273–279. doi:10.2307/2032354. JSTOR 2032354. MR 0069174.
  9. ^ Howie, James (1980). "On locally indicable groups". Mathematische Zeitschrift. 182 (4): 445–461. doi:10.1007/BF01214717. MR 0667000. S2CID 121292137.
  10. ^ a b Magnus, Karrass and Solitar, Theorem 4.14, p. 274
  11. ^ Lyndon & Schupp, Ch. II, Section 5, Proposition 5.11
  12. ^ Murasugi, Kunio (1964). "The center of a group with a single defining relation". Mathematische Annalen. 155 (3): 246–251. doi:10.1007/BF01344162. MR 0163945. S2CID 119454184.
  13. ^ Magnus, Wilhelm (1931). "Untersuchungen über einige unendliche diskontinuierliche Gruppen". Mathematische Annalen. 105 (1): 52–74. doi:10.1007/BF01455808. MR 1512704. S2CID 120949491.
  14. ^ Lyndon & Schupp, p. 112
  15. ^ Gilbert Baumslag; Donald Solitar (1962). "Some two-generator one-relator non-Hopfian groups". Bulletin of the American Mathematical Society. 68 (3): 199–201. doi:10.1090/S0002-9904-1962-10745-9. MR 0142635.
  16. ^ Chebotarʹ, A.A. (1971). "Subgroups of groups with one defining relation that do not contain free subgroups of rank 2" (PDF). Algebra i Logika. 10 (5): 570–586. MR 0313404.
  17. ^ Cohen, Daniel E.; Lyndon, Roger C. (1963). "Free bases for normal subgroups of free groups". Transactions of the American Mathematical Society. 108 (3): 526–537. doi:10.1090/S0002-9947-1963-0170930-9. MR 0170930.
  18. ^ Karrass, A.; Magnus, W.; Solitar, D. (1960). "Elements of finite order in groups with a single defining relation". Communications on Pure and Applied Mathematics. 13: 57–66. doi:10.1002/cpa.3160130107. MR 0124384.
  19. ^ a b Newman, B. B. (1968). "Some results on one-relator groups". Bulletin of the American Mathematical Society. 74 (3): 568–571. doi:10.1090/S0002-9904-1968-12012-9. MR 0222152.
  20. ^ Lyndon & Schupp, Ch. IV, Theorem 5.5, p. 205
  21. ^ Howie, James (1984). "Cohomology of one-relator products of locally indicable groups". Journal of the London Mathematical Society. 30 (3): 419–430. doi:10.1112/jlms/s2-30.3.419. MR 0810951.
  22. ^ a b Baumslag, Gilbert; Fine, Benjamin; Rosenberger, Gerhard (2019). "One-relator groups: an overview". Groups St Andrews 2017 in Birmingham. London Math. Soc. Lecture Note Ser. Vol. 455. Cambridge University Press. pp. 119–157. ISBN 978-1-108-72874-4. MR 3931411.
  23. ^ Louder, Larsen; Wilton, Henry (2020). "One-relator groups with torsion are coherent". Mathematical Research Letters. 27 (5): 1499–1512. arXiv:1805.11976. doi:10.4310/MRL.2020.v27.n5.a9. MR 4216595. S2CID 119141737.
  24. ^ Dahmani, Francois; Guirardel, Vincent (2011). "The isomorphism problem for all hyperbolic groups". Geometric and Functional Analysis. 21 (2): 223–300. arXiv:1002.2590. doi:10.1007/s00039-011-0120-0. MR 2795509.
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  26. ^ Kielak, Dawid; Linton, Marco (2024). "Virtually free-by-cyclic groups". Geometric and Functional Analysis. 34: 1580–1608. doi:10.1007/s00039-024-00687-6. MR 4792841.
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  28. ^ Magnus, Karrass, and Solitar, Section 4.4
  29. ^ Lyndon& Schupp, Chapter IV, Section 5, pp. 198-205
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  31. ^ Brown, Kenneth S. (1987). "Trees, valuations, and the Bieri-Neumann-Strebel invariant". Inventiones Mathematicae. 90 (3): 479–504. Bibcode:1987InMat..90..479B. doi:10.1007/BF01389176. MR 0914847. S2CID 122703100., Theorem 4.3
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  33. ^ Gersten, S. M. (1987). "Nonsingular equations of small weight over groups". Combinatorial group theory and topology (Alta, Utah, 1984). Annals of Mathematics Studies. Vol. 111. Princeton University Press. pp. 121–144. doi:10.1515/9781400882083-007. ISBN 0-691-08409-2. MR 0895612.
  34. ^ Klyachko, A. A. (1993). "A funny property of sphere and equations over groups". Communications in Algebra. 21 (7): 2555–2575. doi:10.1080/00927879308824692. MR 1218513.
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