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.
A one-relator group is a group G that admits a group presentation of the form
(1) |
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 ( 1) is usually assumed to be finite where one-relator groups are discussed, in which case ( 1) can be rewritten more explicitly as
(2) |
where for some integer
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 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, [1] known as Freiheitssatz, states that in this situation H is freely generated by , that is, . See also [2] [3] for other proofs.
Here we assume that a one-relator group G is given by presentation ( 2) with a finite generating set and a nontrivial freely and cyclically reduced defining relation .
Suppose a one-relator group G given by presentation ( 2) where where and where is not a proper power (and thus s is also freely and cyclically reduced). Then the following hold:
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 (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 in this case. For every generator one denotes where . Then r can be rewritten as a word in these new generators with .
For example, if then .
Let be the alphabet consisting of the portion of given by all with where are the minimum and the maximum subscripts with which occurs in .
Magnus observed that the subgroup is itself a one-relator group with the one-relator presentation . Note that since , one can usually apply the inductive hypothesis to when proving a particular statement about G.
Moreover, if for then is also a one-relator group, where is obtained from by shifting all subscripts by . Then the normal closure of in G is
Magnus' original approach exploited the fact that N is actually an iterated amalgamated product of the groups , 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 its minimum and maximum subscripts in are equal then and the inductive step is usually easy to handle in this case.
Suppose then that some generator from occurs in with at least two distinct subscripts. We put to be the set of all generators from with non-maximal subscripts and we put to be the set of all generators from with non-maximal subscripts. (Hence every generator from and from occurs in with a non-unique subscript.) Then and are free Magnus subgroups of L and . Moldavansky observed that in this situation
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 occur in r with nonzero exponents accordingly. Consider a homomorphism given by and fixing the other generators from X. Then for the exponent sum on y is equal to 0. The map f induces a group homomorphism that turns out to be an embedding. The one-relator group G' can then be treated using Moldavansky's approach. When splits as an HNN-extension of a one-relator group L, the defining relator 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.
It turns out that many two-generator one-relator groups split as semidirect products . 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 and let be an epimorphism. One can then change a free basis of to a basis such that and rewrite the presentation of G in this generators as
where is a freely and cyclically reduced word.
Since , the exponent sum on t in r is equal to 0. Again putting , we can rewrite r as a word in Let be the minimum and the maximum subscripts of the generators occurring in . Brown showed [30] that is finitely generated if and only if and both and occur exactly once in , and moreover, in that case the group is free. Therefore if is an epimorphism with a finitely generated kernel, then G splits as where is a finite rank free group.
Later Dunfield and Thurston proved [31] that if a one-relator two-generator group is chosen "at random" (that is, a cyclically reduced word r of length n in is chosen uniformly at random) then the probability that a homomorphism from G onto with a finitely generated kernel exists satisfies
for all sufficiently large n. Moreover, their experimental data indicates that the limiting value for is close to .
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.
A one-relator group is a group G that admits a group presentation of the form
(1) |
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 ( 1) is usually assumed to be finite where one-relator groups are discussed, in which case ( 1) can be rewritten more explicitly as
(2) |
where for some integer
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 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, [1] known as Freiheitssatz, states that in this situation H is freely generated by , that is, . See also [2] [3] for other proofs.
Here we assume that a one-relator group G is given by presentation ( 2) with a finite generating set and a nontrivial freely and cyclically reduced defining relation .
Suppose a one-relator group G given by presentation ( 2) where where and where is not a proper power (and thus s is also freely and cyclically reduced). Then the following hold:
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 (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 in this case. For every generator one denotes where . Then r can be rewritten as a word in these new generators with .
For example, if then .
Let be the alphabet consisting of the portion of given by all with where are the minimum and the maximum subscripts with which occurs in .
Magnus observed that the subgroup is itself a one-relator group with the one-relator presentation . Note that since , one can usually apply the inductive hypothesis to when proving a particular statement about G.
Moreover, if for then is also a one-relator group, where is obtained from by shifting all subscripts by . Then the normal closure of in G is
Magnus' original approach exploited the fact that N is actually an iterated amalgamated product of the groups , 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 its minimum and maximum subscripts in are equal then and the inductive step is usually easy to handle in this case.
Suppose then that some generator from occurs in with at least two distinct subscripts. We put to be the set of all generators from with non-maximal subscripts and we put to be the set of all generators from with non-maximal subscripts. (Hence every generator from and from occurs in with a non-unique subscript.) Then and are free Magnus subgroups of L and . Moldavansky observed that in this situation
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 occur in r with nonzero exponents accordingly. Consider a homomorphism given by and fixing the other generators from X. Then for the exponent sum on y is equal to 0. The map f induces a group homomorphism that turns out to be an embedding. The one-relator group G' can then be treated using Moldavansky's approach. When splits as an HNN-extension of a one-relator group L, the defining relator 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.
It turns out that many two-generator one-relator groups split as semidirect products . 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 and let be an epimorphism. One can then change a free basis of to a basis such that and rewrite the presentation of G in this generators as
where is a freely and cyclically reduced word.
Since , the exponent sum on t in r is equal to 0. Again putting , we can rewrite r as a word in Let be the minimum and the maximum subscripts of the generators occurring in . Brown showed [30] that is finitely generated if and only if and both and occur exactly once in , and moreover, in that case the group is free. Therefore if is an epimorphism with a finitely generated kernel, then G splits as where is a finite rank free group.
Later Dunfield and Thurston proved [31] that if a one-relator two-generator group is chosen "at random" (that is, a cyclically reduced word r of length n in is chosen uniformly at random) then the probability that a homomorphism from G onto with a finitely generated kernel exists satisfies
for all sufficiently large n. Moreover, their experimental data indicates that the limiting value for is close to .