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''independence theorem''

A theorem originally proposed by M. Dehn in a geometrical setting and originally proven by W. Magnus [[#References|[a1]]]. This theorem is the cornerstone of one-relator [[Group|group]] theory.

The Freiheitssatz says the following: Let $G = \langle x _ { 1 } , \dots , x _ { n } : r = 1 \rangle$ be a [[Group|group]] defined by a single cyclically reduced relator $r$. If $x_{1} $ appears in $r$, then the subgroup of $G$ generated by $x _ { 2 } , \dots , x _ { n }$ is a [[Free group|free group]], freely generated by $x _ { 2 } , \dots , x _ { n }$.

In coarser language, the theorem says that if $G$ is as above, then the only relations in $x _ { 2 } , \dots , x _ { n }$ 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 $V = K ^ { n }$ is a [[Linear space|linear space]] over a field $K$ with a basis $e _ { 1 } , \ldots , e _ { n }$. If $W = \operatorname { lin } ( w )$ is the subspace of $V$ generated by a vector $w = \sum _ { i = 1 } ^ { n } m _ { i } e _ { i }$ with $m _ { 1 } \neq 0$, then the elements $e _ { 2 } , \dots , e _ { n }$ are linearly independent modulo $W$.

Magnus' method of proof of the Freiheitssatz relies on amalgamations (cf. also [[Amalgam|Amalgam]]; [[Amalgam of groups|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|Identity problem]]; [[#References|[a2]]]).

There are two general approaches to extending the Freiheitssatz. The first is concerned with the notion of the one-relator product $G = * A _ { i } /  N ( r )$ of a family $\{ A _ { i } \}$ of groups, where the element $r$ is cyclically reduced and of syllable length at least $2$ and $N ( r )$ is its normal closure in $* A_i$. Some authors (see [[#References|[a3]]]) give conditions for the factors $A_i$ to inject into $G$.

The second approach is concerned with multi-relator versions of the Freiheitssatz (see [[#References|[a3]]] for a list of references). For example, the following strong result by N.S. Romanovskii [[#References|[a4]]] holds: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017027.png"/> have deficiency $d = n - m &gt; 0$. Then there exist a subset of $d$ of the given generators which freely generates a subgroup of $G$.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  W. Magnus,   "Über discontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz)"  ''J. Reine Angew. Math.'' , '''163'''  (1930)  pp. 141–165</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  W. Magnus,   "Das Identitätsproblem für Gruppen mit einer definierenden Relation"  ''Math. Ann.'' , '''106'''  (1932)  pp. 295–307</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  B. Fine,   G. Rosenberger,   "The Freiheitssatz and its extensions"  ''Contemp. Math.'' , '''169'''  (1994)  pp. 213–252</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  N.S. Romanovskii,   "Free subgroups of finitely presented groups"  ''Algebra i Logika'' , '''16'''  (1977)  pp. 88–97  (In Russian)</td></tr></table>