One-relator group explained

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,} ... , x _{n\}} for some integer n\ge 1 .

Freiheitssatz

See 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,...,} 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.

• Every one-relator group G is virtually torsion-free, that is, admits a torsion-free subgroup of finite index.^[4]
• A one-relator presentation is diagrammatically aspherical.^[5]
• If

r\in F(x _{1,\ldots,x} _{n)} is not a proper power then the presentation complex P for presentation is a finite Eilenberg–MacLane complex K(G, 1 ) .^[6]

• If

r\in F(x _{1,\ldots,x} _{n)} is not a proper power then a one-relator group G has cohomological dimension \le 2 .

• 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 , ... ,n either x _{i} or x ^{-1} _{i} 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 Z .^[9]

• Every one-relator group G has algorithmically decidable word problem.^[10]
• If G is a one-relator group and

H\le G is a Magnus subgroup then the subgroup membership problem for H in G is decidable.^[10]

• It is unknown if one-relator groups have solvable conjugacy problem.
• It is unknown if the isomorphism problem is decidable for the class of one-relator groups.
• 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,...,} 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]} B( 2 , 3 ) = \langle a,b\mid b ^{-1} a ^{2b=a} ^{3\rangle} .^[15]

• 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 ^{-1} _{i} 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]

• G admits a torsion-free normal subgroup of finite index.^[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 a word-hyperbolic group.^[22]
• G has decidable conjugacy problem.^[19]
• 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 residually finite.^[25]

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, ... , z\} in this case. For every generator x\in X\setminus \{t\} one denotes x ^{- i} _{i = t} xt ^{i} where i\in Z . Then r can be rewritten as a word r _{0} in these new generators X _{infty} = \{(a _{i)} _{i,} (b _{i)} _{i,} ... , (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 a} _{-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 ^{\pm 1} _{i} 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} _{i = t} X ^{i} _{0 t} for i\in 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 isN = \left\langle c up _{i\in} 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\rtimesZ} . 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 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 asG = \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} _{i = t} at ^{i} , we can rewrite r as a word r _{0} in (a _{i)} _{i\in} . 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 Z is an epimorphism with a finitely generated kernel, then G splits as G = F _{m\rtimes} 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 Z with a finitely generated kernel exists satisfies0 . 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

Z x Z=\langle a, b \mid a ^{-1} b ^{-1} ab =1 \rangle B(m, n) = \langle a,b\mid b ^{-1} a ^{m} b = a ^{n\rangle} where m,n\ne 0 .

• Torus knot group

G = \langle a, b\mid a ^{p=b} ^{q\rangle} where p,q\ge 1 are coprime integers.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 G = \langle a _{1,} b _{1,} ... , a _{n,} b _{n\mid} [a _{1,b} _{1]...} [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,...,} a _{n\mid} a ^{2 …} _{1} a ^{2=1 \rangle} _{n} , 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

• 3-manifolds
• Geometric topology
• Small cancellation theory

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. .
• Book: Lyndon . Roger C. . Roger Lyndon . Schupp . Paul E. . Paul Schupp . 3-540-41158-5 . 1812024 . Springer-Verlag, Berlin . Classics in Mathematics . Combinatorial group theory . 2001.

External links

• Andrew Putman's notes on one-relator groups, University of Notre Dame

Notes and References

1. Magnus . Wilhelm . Über diskontinuierliche Gruppen mit einer definierenden Relation. (Der Freiheitssatz). . . 1930 . 1930 . 163 . 141–165 . 10.1515/crll.1930.163.141. 1581238. 117245586 .
2. Lyndon . Roger C. . Roger Lyndon . 10.1112/jlms/s2-5.1.95 . . 0294465 . 95–101 . Second Series . On the Freiheitssatz . 5 . 1972 . 2027.42/135658 . free .
3. Weinbaum . C. M. . Illinois Journal of Mathematics . 0297849 . 308–322 . On relators and diagrams for groups with one defining relation . 16 . 1972 . 2 . 10.1215/ijm/1256052287 . free .
4. Fischer . J. . Karrass . A. . Solitar . D. . On one-relator groups having elements of finite order . . 1972 . 33 . 2 . 297–301 . 10.2307/2038048. 2038048 . 311780. free.
5. Lyndon & Schupp, Ch. III, Section 11, Proposition 11.1, p. 161
6. Dyer . Eldon . Vasquez . A. T. . Some small aspherical spaces . . 1973 . 16 . 3 . 332–352 . 10.1017/S1446788700015147. 0341476. free .
7. Magnus, Karrass and Solitar, Theorem N3, p. 167
8. Shenitzer . Abe . Decomposition of a group with a single defining relation into a free product . . 1955 . 6 . 2 . 273–279 . 10.2307/2032354. 2032354 . 69174. free.
9. Howie . James . On locally indicable groups . . 1980 . 182 . 4 . 445–461 . 10.1007/BF01214717. 667000. 121292137 .
10. Magnus, Karrass and Solitar, Theorem 4.14, p. 274
11. Lyndon & Schupp, Ch. II, Section 5, Proposition 5.11
12. Murasugi . Kunio . The center of a group with a single defining relation . . 1964 . 155 . 3 . 246–251 . 10.1007/BF01344162. 163945. 119454184 .
13. Magnus . Wilhelm . Untersuchungen über einige unendliche diskontinuierliche Gruppen . . 1931 . 105 . 1 . 52–74 . 10.1007/BF01455808. 1512704. 120949491 .
14. Lyndon & Schupp, p. 112
15. Gilbert Baumslag. Donald Solitar. Some two-generator one-relator non-Hopfian groups. Bulletin of the American Mathematical Society. 68. 1962. 3. 199–201. 0142635. 10.1090/S0002-9904-1962-10745-9. free.
16. Chebotarʹ . A.A. . Subgroups of groups with one defining relation that do not contain free subgroups of rank 2 . . 1971 . 10 . 5 . 570–586 . 0313404.
17. Cohen . Daniel E. . Lyndon . Roger C. . Free bases for normal subgroups of free groups . . 1963 . 108 . 3 . 526–537 . 10.1090/S0002-9947-1963-0170930-9. 0170930. free.
18. Karrass . A. . Magnus . W. . Solitar . D. . Elements of finite order in groups with a single defining relation . . 1960 . 13 . 57–66 . 10.1002/cpa.3160130107. 124384.
19. Newman . B. B. . Some results on one-relator groups . . 1968 . 74 . 3 . 568–571 . 10.1090/S0002-9904-1968-12012-9. 222152. free.
20. Lyndon & Schupp, Ch. IV, Theorem 5.5, p. 205
21. Howie . James . Cohomology of one-relator products of locally indicable groups . . 1984 . 30 . 3 . 419–430 . 10.1112/jlms/s2-30.3.419. 810951.
22. Book: Baumslag . Gilbert . Fine . Benjamin . Rosenberger . Gerhard . Groups St Andrews 2017 in Birmingham . 2019 . Cambridge University Press . 978-1-108-72874-4 . 119–157 . One-relator groups: an overview. https://books.google.com/books?id=mkaPDwAAQBAJ&dq=one-relator+groups+with+torsion+are+word-hyperbolic&pg=PA136. 3931411. London Math. Soc. Lecture Note Ser.. 455.
23. Louder . Larsen . Wilton . Henry . One-relator groups with torsion are coherent . Mathematical Research Letters . 2020 . 27 . 5 . 1499–1512 . 10.4310/MRL.2020.v27.n5.a9. 1805.11976 . 4216595. 119141737 .
24. Dahmani . Francois . Guirardel . Vincent . The isomorphism problem for all hyperbolic groups . . 2011 . 21 . 2 . 223–300 . 10.1007/s00039-011-0120-0. 2795509. free. 1002.2590 .
25. Wise . Daniel T. . Research announcement: the structure of groups with a quasiconvex hierarchy . Electronic Research Announcements in Mathematical Sciences . 2009 . 16 . 44–55 . 10.3934/era.2009.16.44. 2558631. free.
26. Kielak . Dawid . Linton . Marco . Virtually free-by-cyclic groups . Geometric and Functional Analysis . 2024 . 34 . 1580-1608 . 10.1007/s00039-024-00687-6. 4792841. free.
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. . Certain subgroups of groups with one defining relation . . 1967 . 8 . 1370–1384. 10.1007/BF02196411 . 0220810. 119585707 .
31. Brown . Kenneth S. . Trees, valuations, and the Bieri-Neumann-Strebel invariant . . 1987 . 90 . 3 . 479–504 . 10.1007/BF01389176. 1987InMat..90..479B . 0914847. 122703100 ., Theorem 4.3
32. Dunfield . Nathan . Thurston . Dylan . A random tunnel number one 3–manifold does not fiber over the circle . . 2006 . 10 . 4 . 2431–2499 . math/0510129. 10.2140/gt.2006.10.2431. free. 2284062., Theorem 6.1
33. Book: Gersten . S. M. . Combinatorial group theory and topology (Alta, Utah, 1984) . 1987 . Princeton University Press . 0-691-08409-2 . 121–144 . Nonsingular equations of small weight over groups. Annals of Mathematics Studies. 111. 10.1515/9781400882083-007 . 0895612.
34. Klyachko . A. A. . A funny property of sphere and equations over groups . . 1993 . 21 . 7 . 2555–2575 . 10.1080/00927879308824692. 1218513.