One-relator group

• G has virtual cohomological dimension ${\displaystyle \leq 2}$.^[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]

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^[26] and Section 4.4 of Magnus, Karrass and Solitar^[27] for Magnus' original approach and Section 5 of Chapter IV of Lyndon and Schupp^[28] for the Moldavansky's HNN-extension version of that approach.^[29]

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 ${\displaystyle \#X\geq 2}$ (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 ${\displaystyle X=\{t,a,b,\dots ,z\}}$ in this case. For every generator ${\displaystyle x\in X\setminus \{t\}}$ one denotes ${\displaystyle x_{i}=t^{-i}xt^{i}}$ where ${\displaystyle i\in \mathbb {Z} }$. Then r can be rewritten as a word ${\displaystyle r_{0}}$ in these new generators ${\displaystyle X_{\infty }=\{(a_{i})_{i},(b_{i})_{i},\dots ,(z_{i})_{i}\}}$ with ${\displaystyle |r_{0}|<|r|}$.

For example, if ${\displaystyle r=t^{-2}btat^{3}b^{-2}a^{2}t^{-1}at^{-1}}$ then ${\displaystyle r_{0}=b_{2}a_{1}b_{-2}^{-2}a_{-2}^{2}a_{-1}}$.

Let ${\displaystyle X_{0}}$ be the alphabet consisting of the portion of ${\displaystyle X_{\infty }}$ given by all ${\displaystyle x_{i}}$ with ${\displaystyle m(x)\leq i\leq M(x)}$ where ${\displaystyle m(x),M(x)}$ are the minimum and the maximum subscripts with which ${\displaystyle x_{i}^{\pm 1}}$ occurs in ${\displaystyle r_{0}}$.

Magnus observed that the subgroup ${\displaystyle L=\langle X_{0}\rangle \leq G}$ is itself a one-relator group with the one-relator presentation ${\displaystyle L=\langle X_{0}\mid r_{0}=1\rangle }$. Note that since ${\displaystyle |r_{0}|<|r|}$, one can usually apply the inductive hypothesis to ${\displaystyle L}$ when proving a particular statement about G.

Moreover, if ${\displaystyle X_{i}=t^{-i}X_{0}t^{i}}$ for ${\displaystyle i\in \mathbb {Z} }$ then ${\displaystyle L_{i}=\langle X_{i}\rangle =\langle X_{i}|r_{i}=1\rangle }$ is also a one-relator group, where ${\displaystyle r_{i}}$ is obtained from ${\displaystyle r_{0}}$ by shifting all subscripts by ${\displaystyle i}$. Then the normal closure ${\displaystyle N=\langle \langle X_{0}\rangle \rangle _{G}}$ of ${\displaystyle X_{0}}$ in G is

${\displaystyle N=\left\langle \bigcup _{i\in \mathbb {Z} }L_{i}\right\rangle .}$

Magnus' original approach exploited the fact that N is actually an iterated

${\displaystyle L_{i}}$

Later Moldavansky simplified the framework and noted that in this case G itself is an

If for every generator from ${\displaystyle X_{0}}$ its minimum and maximum subscripts in ${\displaystyle r_{0}}$ are equal then ${\displaystyle G=L\ast \langle t\rangle }$ and the inductive step is usually easy to handle in this case.

Suppose then that some generator from ${\displaystyle X_{0}}$ occurs in ${\displaystyle r_{0}}$ with at least two distinct subscripts. We put ${\displaystyle Y_{-}}$ to be the set of all generators from ${\displaystyle X_{0}}$ with non-maximal subscripts and we put ${\displaystyle Y_{+}}$ to be the set of all generators from ${\displaystyle X_{0}}$ with non-maximal subscripts. (Hence every generator from ${\displaystyle Y_{-}}$ and from ${\displaystyle Y_{-}}$ occurs in ${\displaystyle r_{0}}$ with a non-unique subscript.) Then ${\displaystyle H_{-}=\langle Y_{-}\rangle }$ and ${\displaystyle H_{+}=\langle Y_{+}\rangle }$ are free Magnus subgroups of L and ${\displaystyle t^{-1}H_{-}t=H_{+}}$. Moldavansky observed that in this situation

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

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

${\displaystyle G=\langle a,t\mid r=1\rangle }$

where ${\displaystyle 1\neq r=r(a,t)\in F(a,t)}$ is a freely and cyclically reduced word.

Since ${\displaystyle \phi (r)=0,\phi (t)=1}$, the exponent sum on t in r is equal to 0. Again putting ${\displaystyle a_{i}=t^{-i}at^{i}}$, we can rewrite r as a word ${\displaystyle r_{0}}$ in ${\displaystyle (a_{i})_{i\in \mathbb {Z} }.}$ Let ${\displaystyle m,M}$ be the minimum and the maximum subscripts of the generators occurring in ${\displaystyle r_{0}}$. Brown showed^[30] that ${\displaystyle \ker(\phi )}$ is finitely generated if and only if ${\displaystyle m<M}$ and both ${\displaystyle a_{m}}$ and ${\displaystyle a_{M}}$ occur exactly once in ${\displaystyle r_{0}}$, and moreover, in that case the group ${\displaystyle \ker(\phi )}$ is free. Therefore if ${\displaystyle \phi :G\to \mathbb {Z} }$ is an epimorphism with a finitely generated kernel, then G splits as ${\displaystyle G=F_{m}\rtimes \mathbb {Z} }$ where ${\displaystyle F_{m}=\ker(\phi )}$ is a finite rank free group.

Later Dunfield and Thurston proved^[31] that if a one-relator two-generator group ${\displaystyle G=\langle x_{1},x_{2}\mid r=1\rangle }$ is chosen "at random" (that is, a cyclically reduced word r of length n in ${\displaystyle F(x_{1},x_{2})}$ is chosen uniformly at random) then the probability ${\displaystyle p_{n}}$ that a homomorphism from G onto ${\displaystyle \mathbb {Z} }$ with a finitely generated kernel exists satisfies

${\displaystyle 0.0006<p_{n}<0.975}$

for all sufficiently large n. Moreover, their experimental data indicates that the limiting value for ${\displaystyle p_{n}}$ is close to ${\displaystyle 0.94}$.

Examples of one-relator groups

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

• Baumslag–Solitar group ${\displaystyle B(m,n)=\langle a,b\mid b^{-1}a^{m}b=a^{n}\rangle }$ where ${\displaystyle m,n\neq 0}$.

• Torus knot group ${\displaystyle G=\langle a,b\mid a^{p}=b^{q}\rangle }$ where ${\displaystyle p,q\geq 1}$ are coprime integers.

• Baumslag–Gersten group ${\displaystyle G=\langle a,t\mid a^{a^{t}}=a^{2}\rangle =\langle a,t\mid (t^{-1}a^{-1}t)a(t^{-1}at)=a^{2}\rangle }$

• Oriented
  surface group
  ${\displaystyle G=\langle a_{1},b_{1},\dots ,a_{n},b_{n}\mid [a_{1},b_{1}]\dots [a_{n},b_{n}]=1\rangle }$ where ${\displaystyle [a,b]=a^{-1}b^{-1}ab}$ and where ${\displaystyle n\geq 1}$.

• Non-oriented surface group ${\displaystyle G=\langle a_{1},\dots ,a_{n}\mid a_{1}^{2}\cdots a_{n}^{2}=1\rangle }$, where ${\displaystyle n\geq 1}$.

Generalizations and open problems

• If A and B are two groups, and ${\displaystyle r\in A\ast B}$ is an element in their free product, one can consider a one-relator product ${\displaystyle G=A\ast B/\langle \langle r\rangle \rangle =\langle A,B\mid r=1\rangle }$.

• The so-called Kervaire conjecture, also known as Kervaire–Laudenbach conjecture, asks if it is true that if A is a nontrivial group and ${\displaystyle B=\langle t\rangle }$ is infinite cyclic then for every ${\displaystyle r\in A\ast B}$ the one-relator product ${\displaystyle G=\langle A,t\mid r=1\rangle }$ is nontrivial.^[32]

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

• 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.

• If G is a finitely generated one-relator group (with or without torsion), ${\displaystyle H\leq G}$ is a torsion-free subgroup of finite index and ${\displaystyle \phi :H\to \mathbb {Z} }$ is an epimorphism then ${\displaystyle \ker(\phi )}$ has cohomological dimension 1 and therefore, by a result of Stallings, is locally free.^[34] Baumslag, with co-authors, showed that in many cases, by a suitable choice of H and ${\displaystyle \phi }$ one can prove that that ${\displaystyle \ker(\phi )}$ is actually free (of infinite rank).^[35][36] These results led to a conjecture^[22] that every finitely generated one-relator group with torsion is virtually free-by-cyclic.

See also

• 3-manifolds
• Geometric topology
• Small cancellation theory

Sources

• MR2109550

• MR 1812024
  .