Freiheitssatz
independence theorem
A theorem originally proposed by M. Dehn in a geometrical setting and originally proven by W. Magnus [a1]. This theorem is the cornerstone of one-relator group theory.
The Freiheitssatz says the following: Let be a group defined by a single cyclically reduced relator
. If
appears in
, then the subgroup of
generated by
is a free group, freely generated by
.
In coarser language, the theorem says that if is as above, then the only relations in
are the trivial ones.
The Freiheitssatz can be considered as a non-commutative analogue of certain more transparent results in commutative algebra. For example, suppose that is a linear space over a field
with a basis
. If
is the subspace of
generated by a vector
with
, then the elements
are linearly independent modulo
.
Magnus' method of proof of the Freiheitssatz relies on amalgamations (cf. also Amalgam; Amalgam of groups). This method initiated the use of these products in the study of infinite discrete groups.
One of the by-products of Magnus' proof was an extraordinary description of the structure of these groups, which allowed him to deduce that one-relator groups have solvable word problem (cf. also Identity problem; [a2]).
There are two general approaches to extending the Freiheitssatz. The first is concerned with the notion of the one-relator product of a family
of groups, where the element
is cyclically reduced and of syllable length at least
and
is its normal closure in
. Some authors (see [a3]) give conditions for the factors
to inject into
.
The second approach is concerned with multi-relator versions of the Freiheitssatz (see [a3] for a list of references). For example, the following strong result by N.S. Romanovskii [a4] holds: Let have deficiency
. Then there exist a subset of
of the given generators which freely generates a subgroup of
.
References
[a1] | W. Magnus, "Über discontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz)" J. Reine Angew. Math. , 163 (1930) pp. 141–165 |
[a2] | W. Magnus, "Das Identitätsproblem für Gruppen mit einer definierenden Relation" Math. Ann. , 106 (1932) pp. 295–307 |
[a3] | B. Fine, G. Rosenberger, "The Freiheitssatz and its extensions" Contemp. Math. , 169 (1994) pp. 213–252 |
[a4] | N.S. Romanovskii, "Free subgroups of finitely presented groups" Algebra i Logika , 16 (1977) pp. 88–97 (In Russian) |
Freiheitssatz. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Freiheitssatz&oldid=50063