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

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

If Y is the set of all letters x\in X that appear in r and X'=X\setminus Y then

G=\langle Y\mid r=1\, \rangle \ast F(X').

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

where X=\{x_1, \dots, x_n\} for some integer n\ge 1.

Freiheitssatz

Main article: Freiheitssatz

Let G be a one-relator group given by presentation () above. Recall that r is a freely and cyclically reduced word in F(X). Let y\in X be a letter such that y or y^{-1} appears in r. Let X_1\subseteq X\setminus \{y\}. The subgroup H=\langle X_1\rangle\le G 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 X_1, that is, H=F(X_1). 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 () with a finite generating set X=\{x_1,\dots, x_n\} and a nontrivial freely and cyclically reduced defining relation 1\ne r\in F(X).

  • A one-relator group G is torsion-free if and only if r\in F(x_1,\ldots,x_n) is not a proper power.
  • A one-relator presentation is diagrammatically aspherical.[5]
  • A one-relator group G is free if and only if r\in F(x_1,\ldots,x_n) is a primitive element; in this case G is free of rank n − 1.[7]
  • Suppose the element r\in F(x_1,\ldots,x_n) is of minimal length under the action of \operatorname{Aut}(F_n), and suppose that for every i=1,\dots,n either x_i or x_i^{-1} occurs in r. Then the group G is freely indecomposable.[8]
  • If r\in F(x_1,\ldots,x_n) 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 \mathbb Z.[9]
  • A one-relator group G given by presentation () has rank n (that is, it cannot be generated by fewer than n elements) unless r\in F(x_1,\ldots,x_n) is a primitive element.[11]
  • Let G be a one-relator group given by presentation (). If n\ge 3 then the center of G is trivial, Z(G)=\{1\}. If n=2 and G is non-abelian with non-trivial center, then the center of G is infinite cyclic.[12]
  • Let r,s\in F(X) where X=\{x_1,\dots, x_n\}. Let N_1=\langle\langle r\rangle\rangle_{F(X)} and N_2=\langle\langle s\rangle\rangle_{F(X)} be the normal closures of r and s in F(X) accordingly. Then N_1=N_2 if and only if r is conjugate to s or s^{-1} in F(X).[13][14]
  • Let G be a one-relator group given by presentation (). 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 (). Then the normal subgroup N=\langle\langle r\rangle\rangle_{F(X)}\le F(X) admits a free basis of the form \{u_i^{-1}ru_i\mid i\in I\} for some family of elements \{u_i\in F(X)\mid i\in I\}.[17]

One-relator groups with torsion

Suppose a one-relator group G given by presentation () where r=s^m where m\ge 2 and where 1\ne s\in F(X) 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 \langle s\rangle in G. Moreover, the subgroup of G generated by all torsion elements is a free product of a family of conjugates of \langle s\rangle in G.[4]
  • Newman's "spelling theorem"[19][20] Let 1\ne w\in F(X) be a freely reduced word such that w=1 in G. Then w contains a subword v such that v is also a subword of r or r^{-1} of length |v|=1+(m-1)|s|. Since m\ge 2 that means that |v|>|r|/2 and presentation () of G is a Dehn presentation.
  • G has virtual cohomological dimension \le 2.[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]
  • G is virtually free-by-cyclic, i.e. G has a subgroup H of finite-index such that there is a free normal subgroup F \triangleleft H with cyclic quotient F/H.[26]

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[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 () with a finite generating set X. Assume also that every generator from X actually occurs in r.

One can usually assume that \#X\ge 2 (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 X=\{t, a, b,\dots, z\} in this case. For every generator x\in X\setminus \{t\} one denotes x_i=t^{-i}xt^i where i\in \mathbb Z. Then r can be rewritten as a word r_0 in these new generators X_{\infty}= \{(a_i)_i, (b_i)_i, \dots, (z_i)_i\} with |r_0|<|r|.

For example, if r=t^{-2}bt a t^3b^{-2}a^2t^{-1}at^{-1} then r_0=b_2a_1b_{-2}^{-2}a_{-2}^2a_{-1}.

Let X_0 be the alphabet consisting of the portion of X_{\infty} given by all x_i with m(x)\le i\le M(x) where m(x), M(x) are the minimum and the maximum subscripts with which x_i^{\pm 1} occurs in r_0.

Magnus observed that the subgroup L=\langle X_0\rangle \le G is itself a one-relator group with the one-relator presentation L=\langle X_0\mid r_0=1\rangle. Note that since |r_0|<|r|, one can usually apply the inductive hypothesis to L when proving a particular statement about G.

Moreover, if X_i=t^{-i}X_0t^i for i\in \mathbb Z then L_i=\langle X_i\rangle=\langle X_i| r_i=1\rangle is also a one-relator group, where r_i is obtained from r_0 by shifting all subscripts by i. Then the normal closure N=\langle \langle X_0\rangle\rangle_G of X_0 in G is

N=\left\langle \bigcup_{i\in \mathbb Z} L_i \right\rangle.

Magnus' original approach exploited the fact that N is actually an iterated amalgamated product of the groups L_i, 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 X_0 its minimum and maximum subscripts in r_0 are equal then G=L\ast \langle t\rangle and the inductive step is usually easy to handle in this case.

Suppose then that some generator from X_0 occurs in r_0 with at least two distinct subscripts. We put Y_- to be the set of all generators from X_0 with non-maximal subscripts and we put Y_+ to be the set of all generators from X_0 with non-maximal subscripts. (Hence every generator from Y_- and from Y_- occurs in r_0 with a non-unique subscript.) Then H_-=\langle Y_-\rangle and H_+=\langle Y_+\rangle are free Magnus subgroups of L and t^{-1}H_- t=H_+. Moldavansky observed that in this situation

G=\langle L, t\mid t^{-1}H_- t=H_+\rangle

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 x,y\in X occur in r with nonzero exponents \alpha, \beta accordingly. Consider a homomorphism f:F(X)\to F(X) given by f(x)=xy^{-\beta}, f(y)=y^\alpha and fixing the other generators from X. Then for r'=f(r)\in F(X) the exponent sum on y is equal to 0. The map f induces a group homomorphism \phi: G\to G'=\langle X\mid r'=1\rangle that turns out to be an embedding. The one-relator group G' can then be treated using Moldavansky's approach. When G' splits as an HNN-extension of a one-relator group L, the defining relator r_0 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 G=F_m\rtimes\mathbb Z. 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 () with n=2 and let \phi:G\to \mathbb Z be an epimorphism. One can then change a free basis of F(X) to a basis t,a such that \phi(t)=1,\phi(a)=0 and rewrite the presentation of G in this generators as

G=\langle a,t\mid r=1\rangle

where 1\ne r=r(a,t)\in F(a,t) is a freely and cyclically reduced word.

Since \phi(r)=0, \phi(t)=1, the exponent sum on t in r is equal to 0. Again putting a_i=t^{-i}at^i, we can rewrite r as a word r_0 in (a_i)_{i\in \mathbb Z}. Let m,M be the minimum and the maximum subscripts of the generators occurring in r_0. Brown showed[31] that \ker(\phi) is finitely generated if and only if m<M and both a_m and a_{M} occur exactly once in r_0, and moreover, in that case the group \ker(\phi) is free. Therefore if \phi:G\to \mathbb Z is an epimorphism with a finitely generated kernel, then G splits as G=F_m\rtimes \mathbb Z where F_m=\ker(\phi) is a finite rank free group.

Later Dunfield and Thurston proved[32] that if a one-relator two-generator group G=\langle x_1,x_2\mid r=1\rangle is chosen "at random" (that is, a cyclically reduced word r of length n in F(x_1,x_2) is chosen uniformly at random) then the probability p_n that a homomorphism from G onto \mathbb Z with a finitely generated kernel exists satisfies

0.0006<p_n<0.975

for all sufficiently large n. Moreover, their experimental data indicates that the limiting value for p_n is close to 0.94.

Examples of one-relator groups

  • Baumslag–Gersten group G=\langle a,t \mid a^{a^t}=a^2\rangle =\langle a, t \mid (t^{-1}a^{-1}t) a (t^{-1} at)=a^2 \rangle
  • Oriented surface group G=\langle a_1, b_1, \dots, a_n, b_n\mid [a_1,b_1]\dots [a_n,b_n]=1\rangle where [a,b]=a^{-1}b^{-1}ab and where n\ge 1.
  • Non-oriented surface group G=\langle a_1,\dots, a_n\mid a_1^2\cdots a_n^2=1\rangle, where n\ge 1.

Generalizations and open problems

  • If A and B are two groups, and r\in A\ast B is an element in their free product, one can consider a one-relator product G=A\ast B/\langle\langle r\rangle\rangle=\langle A, B\mid r=1\rangle.
  • The so-called Kervaire conjecture, also known as Kervaire–Laudenbach conjecture, asks if it is true that if A is a nontrivial group and B=\langle t\rangle is infinite cyclic then for every r\in A\ast B the one-relator product G=\langle A, t\mid r=1\rangle 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

Sources

  • Lyndon, Roger C. & Schupp, Paul E. (2001). Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin. ISBN 3-540-41158-5. MR 1812024.

References

  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. 5: 95–101. Second Series. 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. ^ 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 311780.
  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 69174.
  9. ^ Howie, James (1980). "On locally indicable groups". Mathematische Zeitschrift. 182 (4): 445–461. doi:10.1007/BF01214717. MR 667000. S2CID 121292137
  10. ^ 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 163945. 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". 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 124384.
  19. ^ 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 222152.
  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 810951.
  22. ^ Baumslag, Gilbert; Fine, Benjamin; Rosenberger, Gerhard (2019). "One-relator groups: an overview". Groups St Andrews 2017 in Birmingham. Vol. 455. London Math. Soc. Lecture Note Ser. 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.
  25. ^ Wise, Daniel T. (2009). "Research announcement: the structure of groups with a quasiconvex hierarchy". Electronic Research Announcements in Mathematical Sciences. 16: 44–55. doi:10.3934/era.2009.16.44. MR 2558631.
  26. ^ Kielak, Dawid & Linton, Marco (2024). "Virtually free-by-cyclic groups". Geometric and Functional Analysis. 34: 1580-1608. arXiv:2302.11500. doi:10.1007/s00039-024-00687-6. MR 4792841.
  27. ^ Lyndon& Schupp, Chapter II, Section 6, pp. 111-113
  28. ^ Magnus, Karrass, and Solitar, Section 4.4
  29. ^ Lyndon& Schupp, Chapter IV, Section 5, pp. 198-205
  30. ^ Moldavanskii, D.I. (1967). "Certain subgroups of groups with one defining relation". Siberian Mathematical Journal. 8: 1370–1384. doi:10.1007/BF02196411. MR 0220810. S2CID 119585707
  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
  32. ^ Dunfield, Nathan & Thurston, Dylan (2006). "A random tunnel number one 3–manifold does not fiber over the circle". Geometry & Topology. 10 (4): 2431–2499. arXiv:math/0510129. doi:10.2140/gt.2006.10.2431. MR 2284062., Theorem 6.1
  33. ^ Gersten, S. M. (1987). "Nonsingular equations of small weight over groups". Combinatorial group theory and topology (Alta, Utah, 1984). Vol. 111. Annals of Mathematics Studies. 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.