Namespaces
Variants
Actions

Difference between revisions of "Freiheitssatz"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (AUTOMATIC EDIT (latexlist): Replaced 29 formulas out of 30 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
Line 1: Line 1:
 +
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
 +
 +
Out of 30 formulas, 29 were replaced by TEX code.-->
 +
 +
{{TEX|semi-auto}}{{TEX|partial}}
 
''independence theorem''
 
''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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f1201701.png" /> be a [[Group|group]] defined by a single cyclically reduced relator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f1201702.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f1201703.png" /> appears in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f1201704.png" />, then the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f1201705.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f1201706.png" /> is a [[Free group|free group]], freely generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f1201707.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f1201708.png" /> is as above, then the only relations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f1201709.png" /> are the trivial ones.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017010.png" /> is a [[Linear space|linear space]] over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017011.png" /> with a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017013.png" /> is the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017014.png" /> generated by a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017016.png" />, then the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017017.png" /> are linearly independent modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017018.png" />.
+
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.
 
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.
Line 13: Line 21:
 
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]]]).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017019.png" /> of a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017020.png" /> of groups, where the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017021.png" /> is cyclically reduced and of syllable length at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017023.png" /> is its normal closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017024.png" />. Some authors (see [[#References|[a3]]]) give conditions for the factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017025.png" /> to inject into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017026.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017028.png" />. Then there exist a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017029.png" /> of the given generators which freely generates a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120170/f12017030.png" />.
+
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====
 
====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>
+
<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>

Revision as of 16:55, 1 July 2020

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 $G = \langle x _ { 1 } , \dots , x _ { n } : r = 1 \rangle$ be a 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, 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 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 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 $G = * A _ { i } / N ( r )$ 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 [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 [a3] for a list of references). For example, the following strong result by N.S. Romanovskii [a4] holds: Let have deficiency $d = n - m > 0$. Then there exist a subset of $d$ of the given generators which freely generates a subgroup of $G$.

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)
How to Cite This Entry:
Freiheitssatz. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Freiheitssatz&oldid=16698
This article was adapted from an original article by V.A. Roman'kov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article