# 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 (1) is usually assumed to be finite where one-relator groups are discussed, in which case (1) can be rewritten more explicitly as
where $X=\{x_{1},\dots ,x_{n}\}$ for some integer $n\geq 1.$

## 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 $y\in X$ be a letter such that $y$ or $y^{-1}$ appears in r. Let $X_{1}\subseteq X\setminus \{y\}$. The subgroup $H=\langle X_{1}\rangle \leq G$ 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 (2) with a finite generating set $X=\{x_{1},\dots ,x_{n}\}$ and a nontrivial freely and cyclically reduced defining relation $1\neq r\in F(X)$.
* 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 (2) 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 $\leq 2$.
* 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 $H\leq G$ 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 (2) 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 (2). If $n\geq 3$ 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 $B(2,3)=\langle a,b\mid b^{-1}a^{2}b=a^{3}\rangle$.
* 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.
* Let G be a one-relator group given by presentation (2). Then the normal subgroup $N=\langle \langle r\rangle \rangle _{F(X)}\leq F(X)$ 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 (2) where $r=s^{m}$ where $m\geq 2$ and where $1\neq s\in F(X)$ 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 $1\neq w\in F(X)$ 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 $m\geq 2$ that means that $|v|>|r|/2$ and presentation (2) of G is a Dehn presentation.
* G has virtual cohomological dimension $\leq 2$.
* 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.
* $G$ 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 $F/H$.

## 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 (1) with a finite generating set X. Assume also that every generator from X actually occurs in r.
One can usually assume that $\#X\geq 2$ (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 $r=t^{-2}btat^{3}b^{-2}a^{2}t^{-1}at^{-1}$ then $r_{0}=b_{2}a_{1}b_{-2}^{-2}a_{-2}^{2}a_{-1}$.
Let $X_{0}$ be the alphabet consisting of the portion of $X_{\infty }$ given by all $x_{i}$ with $m(x)\leq i\leq M(x)$ 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 $L=\langle X_{0}\rangle \leq G$ 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 $X_{i}=t^{-i}X_{0}t^{i}$ 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 (2) 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 $1\neq r=r(a,t)\in F(a,t)$ 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 $m,n\neq 0$.
* Torus knot group $G=\langle a,b\mid a^{p}=b^{q}\rangle$ where $p,q\geq 1$ 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 $n\geq 1$.
* Non-oriented surface group $G=\langle a_{1},\dots ,a_{n}\mid a_{1}^{2}\cdots a_{n}^{2}=1\rangle$, where $n\geq 1$.

## 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.