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 is a freely and cyclically reduced word.
If Y is the set of all letters that appear in r and then
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 for some integer
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 be a letter such that or appears in r. Let . The subgroup 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 , that is, . 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 and a nontrivial freely and cyclically reduced defining relation .
A one-relator group G is torsion-free if and only if 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 is not a proper power then the presentation complex P for presentation () is a finite Eilenberg–MacLane complex .
If is not a proper power then a one-relator group G has cohomological dimension .
A one-relator group G is free if and only if is a primitive element; in this case G is free of rank n − 1.
Suppose the element is of minimal length under the action of , and suppose that for every either or occurs in r. Then the group G is freely indecomposable.
If 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 .
Every one-relator group G has algorithmically decidable word problem.
If G is a one-relator group and 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 is a primitive element.
Let G be a one-relator group given by presentation (). If then the center of G is trivial, . If and G is non-abelian with non-trivial center, then the center of G is infinite cyclic.
Let where . Let and be the normal closures of r and s in F(X) accordingly. Then if and only if is conjugate to or 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 .
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 admits a free basis of the form for some family of elements .
One-relator groups with torsion
Suppose a one-relator group G given by presentation () where where and where 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 in G. Moreover, the subgroup of G generated by all torsion elements is a free product of a family of conjugates of in G.
G admits a torsion-free normal subgroup of finite index.
Newman's "spelling theorem" Let be a freely reduced word such that in G. Then w contains a subword v such that v is also a subword of or of length . Since that means that and presentation () of G is a Dehn presentation.
G has virtual cohomological dimension .
G is a word-hyperbolic group.
G has decidable conjugacy problem.