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 formula_1 is a freely and cyclically reduced word.
If "Y" is the set of all letters formula_2 that appear in "r" and formula_3 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 formula_5 for some integer formula_6
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 formula_7 be a letter such that formula_8 or formula_9 appears in "r". Let formula_10. The subgroup formula_11 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 formula_12, that is, formula_13. 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 formula_14 and a nontrivial freely and cyclically reduced defining relation formula_15.
One-relator groups with torsion.
Suppose a one-relator group "G" given by presentation () where formula_46 where formula_47 and where formula_48 is not a proper power (and thus "s" is also freely and cyclically reduced). Then the following hold:
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 () with a finite generating set "X". Assume also that every generator from "X" actually occurs in "r".
One can usually assume that formula_59 (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 formula_60 in this case. For every generator formula_61 one denotes formula_62 where formula_63. Then "r" can be rewritten as a word formula_64 in these new generators formula_65 with formula_66.
For example, if formula_67 then formula_68.
Let formula_69 be the alphabet consisting of the portion of formula_70 given by all formula_25 with formula_72 where formula_73 are the minimum and the maximum subscripts with which formula_74 occurs in formula_64.
Magnus observed that the subgroup formula_76 is itself a one-relator group with the one-relator presentation formula_77. Note that since formula_66, one can usually apply the inductive hypothesis to formula_79 when proving a particular statement about "G".
Moreover, if formula_80 for formula_63 then formula_82 is also a one-relator group, where formula_83 is obtained from formula_64 by shifting all subscripts by formula_85. Then the normal closure formula_86 of formula_69 in "G" is
Magnus' original approach exploited the fact that "N" is actually an iterated amalgamated product of the groups formula_89, 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 formula_69 its minimum and maximum subscripts in formula_64 are equal then formula_92 and the inductive step is usually easy to handle in this case.
Suppose then that some generator from formula_69 occurs in formula_64 with at least two distinct subscripts. We put formula_95 to be the set of all generators from formula_69 with non-maximal subscripts and we put formula_97 to be the set of all generators from formula_69 with non-maximal subscripts. (Hence every generator from formula_95 and from formula_95 occurs in formula_64 with a non-unique subscript.) Then formula_102 and formula_103 are free Magnus subgroups of "L" and formula_104. 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 formula_106 occur in "r" with nonzero exponents formula_107 accordingly. Consider a homomorphism formula_108 given by formula_109 and fixing the other generators from "X". Then for formula_110 the exponent sum on "y" is equal to 0. The map "f" induces a group homomorphism formula_111 that turns out to be an embedding.
The one-relator group "G"' can then be treated using Moldavansky's approach. When formula_112 splits as an HNN-extension of a one-relator group "L", the defining relator formula_64 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 formula_114. 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 () with formula_33 and let formula_116 be an epimorphism. One can then change a free basis of formula_117 to a basis formula_118 such that formula_119 and rewrite the presentation of "G" in this generators as
where formula_121 is a freely and cyclically reduced word.
Since formula_122, the exponent sum on "t" in "r" is equal to 0. Again putting formula_123, we can rewrite "r" as a word formula_64 in formula_125 Let formula_126 be the minimum and the maximum subscripts of the generators occurring in formula_64. Brown showed that formula_128 is finitely generated if and only if formula_129 and both formula_130 and formula_131 occur exactly once in formula_64, and moreover, in that case the group formula_128 is free.
Therefore if formula_116 is an epimorphism with a finitely generated kernel, then "G" splits as formula_135 where formula_136 is a finite rank free group. 
Later Dunfield and Thurston proved that if a one-relator two-generator group formula_137 is chosen "at random" (that is, a cyclically reduced word "r" of length "n" in formula_138 is chosen uniformly at random) then the probability formula_139 that a homomorphism from "G" onto formula_28 with a finitely generated kernel exists satisfies
for all sufficiently large "n". Moreover, their experimental data indicates that the limiting value for formula_139 is close to formula_143.
External links.
 