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 ∈ F ( X ) {\displaystyle r\in F(X)} is a freely and cyclically reduced word. If Y is the set of all letters x ∈ X {\displaystyle x\in X} that appear in r and X ′ = X ∖ Y {\displaystyle X'=X\setminus Y} then
G = ⟨ Y ∣ r = 1 ⟩ ∗ F ( X ′ ) . {\displaystyle 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 , … , x n } {\displaystyle X=\{x_{1},\dots ,x_{n}\}} for some integer n ≥ 1. {\displaystyle 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 ∈ X {\displaystyle y\in X} be a letter such that y {\displaystyle y} or y − 1 {\displaystyle y^{-1}} appears in r. Let X 1 ⊆ X ∖ { y } {\displaystyle X_{1}\subseteq X\setminus \{y\}} . The subgroup H = ⟨ X 1 ⟩ ≤ G {\displaystyle 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 {\displaystyle X_{1}} , that is, H = F ( X 1 ) {\displaystyle 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 , … , x n } {\displaystyle X=\{x_{1},\dots ,x_{n}\}} and a nontrivial freely and cyclically reduced defining relation 1 ≠ r ∈ F ( X ) {\displaystyle 1\neq r\in F(X)} .
… excerpt ends here. Continue reading the full article.
