One-relator group

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
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, known as Freiheitssatz, states that in this situation H is freely generated by $X_1$, that is, $$H=F(X_1)$$. See also 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.


 * Every one-relator group G is virtually torsion-free, that is, admits a torsion-free subgroup of finite index.


 * A one-relator presentation is diagrammatically aspherical.


 * 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)$$.


 * 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.


 * 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.


 * 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 $$\mathbb Z$$.


 * Every one-relator group G has algorithmically decidable word problem.


 * 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.


 * 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.


 * 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.


 * 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).


 * There exists a finitely generated one-relator group that is not Hopfian and therefore not residually finite, for example the Baumslag–Solitar group $$B(2,3)=\langle a,b\mid b^{-1}a^2b=a^3\rangle$$.


 * 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.


 * 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\}$$.

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.


 * 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.


 * G admits a torsion-free normal subgroup of finite index.


 * Newman's "spelling theorem" 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 $$\le 2$$.


 * G is a word-hyperbolic group.


 * G has decidable conjugacy problem.


 * G is coherent, that is every finitely generated subgroup of G is finitely presentable.


 * The isomorphism problem is decidable for finitely generated one-relator groups with torsion, by virtue of their hyperbolicity.


 * G is residually finite.

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

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 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 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

 * Free abelian group $$ \mathbb Z\times \mathbb Z=\langle a, b \mid a^{-1}b^{-1}ab=1\rangle$$


 * Baumslag–Solitar group $$ B(m, n)=\langle a,b\mid b^{-1} a^m b= a^n\rangle$$ 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.


 * 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.


 * Klyachko proved the Kervaire conjecture for the case where A is torsion-free.


 * A conjecture attributed to Gersten 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), $$H\le G$$ is a torsion-free subgroup of finite index and $$\phi:H\to \mathbb Z$$ is an epimorphism then $$\ker(\phi)$$ has cohomological dimension 1 and therefore, by a result of Stallings, is locally free. Baumslag, with co-authors, showed that in many cases, by a suitable choice of H and $$\phi$$ one can prove that that $$\ker(\phi)$$ is actually free (of infinite rank). These results led to a conjecture that every finitely generated one-relator group with torsion is virtually free-by-cyclic.