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In 1949, G. Higman, B.H. Neumann and H. Neumann [[#References|[a4]]] proved several famous embedding theorems for groups using a construction later called the HNN-extension. The theory of HNN-groups is central to geometric and combinatorial group theory and should be viewed in parallel with amalgamated products (cf. also [[Amalgam of groups|Amalgam of groups]]).
 
In 1949, G. Higman, B.H. Neumann and H. Neumann [[#References|[a4]]] proved several famous embedding theorems for groups using a construction later called the HNN-extension. The theory of HNN-groups is central to geometric and combinatorial group theory and should be viewed in parallel with amalgamated products (cf. also [[Amalgam of groups|Amalgam of groups]]).
  
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==HNN-extensions.==
 
==HNN-extensions.==
Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009020.png" /> is an isomorphism of subgroups of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009022.png" /> is not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009023.png" />. The HNN-extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009024.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009025.png" /> has presentation
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Suppose $\mu : A \rightarrow B$ is an isomorphism of subgroups of a group $G$ and $t$ is not in $G$. The HNN-extension of $G$ with respect to $\mu$ has presentation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009026.png" /></td> </tr></table>
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\begin{equation*} \left\langle G \bigcup \{ t \} : ( \operatorname { ker } ( \tau _ { G } ) ) \bigcup \left\{ t ^ { - 1 } a ^ { - 1 } t \mu ( a ) : \forall a \in A \right\} \right\rangle. \end{equation*}
  
The generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009027.png" /> is called the stable letter, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009028.png" /> the base group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009030.png" /> the associated subgroups of this HNN-extension. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009031.png" />, the HNN-extension is called ascending.
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The generator $t$ is called the stable letter, $G$ the base group and $A$ and $B$ the associated subgroups of this HNN-extension. When $A = G$, the HNN-extension is called ascending.
  
Shorthand notation for the above group is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009032.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009033.png" />.
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Shorthand notation for the above group is $\langle G , t : t ^ { - 1 } A t = B , \mu \rangle$ or $G ^ { * } \mu$.
  
In [[#References|[a4]]] it was shown that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009034.png" /> taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009035.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009036.png" /> is a [[Monomorphism|monomorphism]]. The rest of the normal form theorem for HNN-extensions was proved by J.L. Britton in 1963 [[#References|[a1]]] (Britton's lemma): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009037.png" /> be a sequence of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009038.png" /> and let the letter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009039.png" />, with or without subscripts, denote <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009040.png" />. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009041.png" /> will be called reduced if there is no consecutive subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009042.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009043.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009044.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009045.png" />. For a reduced sequence and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009046.png" />, the element
+
In [[#References|[a4]]] it was shown that the mapping $G \rightarrow G ^ { * } \mu$ taking $g \rightarrow g$ for all $g \in G$ is a [[Monomorphism|monomorphism]]. The rest of the normal form theorem for HNN-extensions was proved by J.L. Britton in 1963 [[#References|[a1]]] (Britton's lemma): Let $g _ { 0 } , \ldots , g _ { n }$ be a sequence of elements of $G$ and let the letter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009039.png"/>, with or without subscripts, denote $\pm 1$. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009041.png"/> will be called reduced if there is no consecutive subsequence $t ^ { - 1 } , g _ { i } , t$ with $g _ { i } \in A$ or $t , g _ { i } , t ^ { - 1 }$ with $g_i \in B$. For a reduced sequence and $n \geq 1$, the element
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009047.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009047.png"/></td> </tr></table>
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009048.png" /> is different from the unit element.
+
of $G _ { \mu } ^ { * }$ is different from the unit element.
  
In the original reference [[#References|[a4]]], the following theorem is proved: Every group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009049.png" /> can be embedded in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009050.png" /> in which all elements of the same order are conjugate (cf. also [[Conjugate elements|Conjugate elements]]). In particular, every torsion-free group can be embedded in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009051.png" /> with only two conjugacy classes. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009052.png" /> is countable, so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009053.png" />. Also, every countable group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009054.png" /> can be embedded in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009055.png" /> generated by two elements of infinite order. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009056.png" /> has an element of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009057.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009058.png" /> does. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009059.png" /> is finitely presentable, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009060.png" />.
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In the original reference [[#References|[a4]]], the following theorem is proved: Every group $G$ can be embedded in a group $G ^ { * }$ in which all elements of the same order are conjugate (cf. also [[Conjugate elements|Conjugate elements]]). In particular, every torsion-free group can be embedded in a group $G ^ { * * }$ with only two conjugacy classes. If $G$ is countable, so is $G ^ { * * }$. Also, every countable group $C$ can be embedded in a group $G$ generated by two elements of infinite order. The group $G$ has an element of finite order $n$ if and only if $C$ does. If $C$ is finitely presentable, then so is $G$.
  
For an excellent account of the history of HNN-extensions, see [[#References|[a2]]]. See [[#References|[a5]]], Chap. IV, for basic results and landmark uses of HNN-extensions, such as: the torsion theorem for HNN-extensions; the Collins conjugacy theorem for HNN-extensions; the construction of finitely-presented non-Hopfian groups (in particular, the [[Baumslag–Solitar group|Baumslag–Solitar group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009061.png" /> is non-Hopfian; cf. also [[Non-Hopf group|Non-Hopf group]]); decompositions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009062.png" />-relator groups; Stallings' classification of finitely-generated groups with more than one end in terms of amalgamated products and HNN-extensions; and Stallings' characterization of bipolar structures on groups.
+
For an excellent account of the history of HNN-extensions, see [[#References|[a2]]]. See [[#References|[a5]]], Chap. IV, for basic results and landmark uses of HNN-extensions, such as: the torsion theorem for HNN-extensions; the Collins conjugacy theorem for HNN-extensions; the construction of finitely-presented non-Hopfian groups (in particular, the [[Baumslag–Solitar group|Baumslag–Solitar group]] $\langle b , t : t ^ { - 1 } b ^ { 2 } t = b ^ { 3 } \rangle$ is non-Hopfian; cf. also [[Non-Hopf group|Non-Hopf group]]); decompositions of $1$-relator groups; Stallings' classification of finitely-generated groups with more than one end in terms of amalgamated products and HNN-extensions; and Stallings' characterization of bipolar structures on groups.
  
 
HNN-extensions are of central importance in, e.g., the modern version of the Van Kampen theorem (based on topological results in [[#References|[a8]]], [[#References|[a9]]]); the Bass–Serre theory of groups acting on trees and the theory of graphs of groups (see [[#References|[a7]]]); Dunwoody's accessibility theorem [[#References|[a3]]]; and JSJ decompositions of groups [[#References|[a6]]].
 
HNN-extensions are of central importance in, e.g., the modern version of the Van Kampen theorem (based on topological results in [[#References|[a8]]], [[#References|[a9]]]); the Bass–Serre theory of groups acting on trees and the theory of graphs of groups (see [[#References|[a7]]]); Dunwoody's accessibility theorem [[#References|[a3]]]; and JSJ decompositions of groups [[#References|[a6]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Britton,  "The word problem"  ''Ann. of Math.'' , '''77'''  (1963)  pp. 16–32</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Chandler,  W. Magnus,  "The history of combinatorial group theory: A case study in the history of ideas" , ''Studies History Math. and Phys. Sci.'' , '''9''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.J. Dunwoody,  "The accessibility of finitely presented groups"  ''Invent. Math.'' , '''81'''  (1985)  pp. 449–457</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Higman,  B.H. Neumann,  H. Neumann,  "Embedding theorems for groups"  ''J. London Math. Soc.'' , '''24'''  (1949)  pp. 247–254; II.4, 13</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Lyndon,  P. Schupp,  "Combinatorial group theory" , Springer  (1977)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Rips,  Z. Sela,  "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition"  ''Ann. of Math. (2)'' , '''146''' :  1  (1997)  pp. 53–109</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.P. Serre,  "Arbres, amalgams, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130090/h13009063.png" />"  ''Astéerisque'' , '''46'''  (1977)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E.R. Van Kampen,  "On the connection between the fundamental groups of some related spaces"  ''Amer. J. Math.'' , '''55'''  (1933)  pp. 261–267</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.R. Van Kampen,  "On some lemmas in the theory of groups"  ''Amer. J. Math.'' , '''55'''  (1933)  pp. 268–273</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  J.L. Britton,  "The word problem"  ''Ann. of Math.'' , '''77'''  (1963)  pp. 16–32</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  B. Chandler,  W. Magnus,  "The history of combinatorial group theory: A case study in the history of ideas" , ''Studies History Math. and Phys. Sci.'' , '''9''' , Springer  (1982)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M.J. Dunwoody,  "The accessibility of finitely presented groups"  ''Invent. Math.'' , '''81'''  (1985)  pp. 449–457</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  G. Higman,  B.H. Neumann,  H. Neumann,  "Embedding theorems for groups"  ''J. London Math. Soc.'' , '''24'''  (1949)  pp. 247–254; II.4, 13</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  R. Lyndon,  P. Schupp,  "Combinatorial group theory" , Springer  (1977)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  E. Rips,  Z. Sela,  "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition"  ''Ann. of Math. (2)'' , '''146''' :  1  (1997)  pp. 53–109</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J.P. Serre,  "Arbres, amalgams, $S L _ { 2 }$"  ''Astéerisque'' , '''46'''  (1977)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  E.R. Van Kampen,  "On the connection between the fundamental groups of some related spaces"  ''Amer. J. Math.'' , '''55'''  (1933)  pp. 261–267</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  E.R. Van Kampen,  "On some lemmas in the theory of groups"  ''Amer. J. Math.'' , '''55'''  (1933)  pp. 268–273</td></tr></table>

Revision as of 16:58, 1 July 2020

In 1949, G. Higman, B.H. Neumann and H. Neumann [a4] proved several famous embedding theorems for groups using a construction later called the HNN-extension. The theory of HNN-groups is central to geometric and combinatorial group theory and should be viewed in parallel with amalgamated products (cf. also Amalgam of groups).

The easiest way to define an HNN-group is in terms of presentations of groups.

Presentation of groups.

A presentation of a group $G$ is a pair $\langle X | R \rangle$ where $R$ is a subset of $F(X)$, the free group on the set $X$, and $G$ is isomorphic (cf. also Isomorphism) to the quotient group $F(X)/N(R)$, where $N(R)$ is the intersection of all normal subgroups of $F(X)$ containing $R$. The subgroup $N(R)$ is called the normal closure of $R$ in $F(X)$.

Given an arbitrary group $G$, there is an obvious homomorphism $\tau_G : F(G) \rightarrow G$ such that $\tau_G(g) = g$ for all $g \in G$. Clearly, $\langle G | \ker \tau_G \rangle$ is a presentation for $G$.

HNN-extensions.

Suppose $\mu : A \rightarrow B$ is an isomorphism of subgroups of a group $G$ and $t$ is not in $G$. The HNN-extension of $G$ with respect to $\mu$ has presentation

\begin{equation*} \left\langle G \bigcup \{ t \} : ( \operatorname { ker } ( \tau _ { G } ) ) \bigcup \left\{ t ^ { - 1 } a ^ { - 1 } t \mu ( a ) : \forall a \in A \right\} \right\rangle. \end{equation*}

The generator $t$ is called the stable letter, $G$ the base group and $A$ and $B$ the associated subgroups of this HNN-extension. When $A = G$, the HNN-extension is called ascending.

Shorthand notation for the above group is $\langle G , t : t ^ { - 1 } A t = B , \mu \rangle$ or $G ^ { * } \mu$.

In [a4] it was shown that the mapping $G \rightarrow G ^ { * } \mu$ taking $g \rightarrow g$ for all $g \in G$ is a monomorphism. The rest of the normal form theorem for HNN-extensions was proved by J.L. Britton in 1963 [a1] (Britton's lemma): Let $g _ { 0 } , \ldots , g _ { n }$ be a sequence of elements of $G$ and let the letter , with or without subscripts, denote $\pm 1$. A sequence will be called reduced if there is no consecutive subsequence $t ^ { - 1 } , g _ { i } , t$ with $g _ { i } \in A$ or $t , g _ { i } , t ^ { - 1 }$ with $g_i \in B$. For a reduced sequence and $n \geq 1$, the element

of $G _ { \mu } ^ { * }$ is different from the unit element.

In the original reference [a4], the following theorem is proved: Every group $G$ can be embedded in a group $G ^ { * }$ in which all elements of the same order are conjugate (cf. also Conjugate elements). In particular, every torsion-free group can be embedded in a group $G ^ { * * }$ with only two conjugacy classes. If $G$ is countable, so is $G ^ { * * }$. Also, every countable group $C$ can be embedded in a group $G$ generated by two elements of infinite order. The group $G$ has an element of finite order $n$ if and only if $C$ does. If $C$ is finitely presentable, then so is $G$.

For an excellent account of the history of HNN-extensions, see [a2]. See [a5], Chap. IV, for basic results and landmark uses of HNN-extensions, such as: the torsion theorem for HNN-extensions; the Collins conjugacy theorem for HNN-extensions; the construction of finitely-presented non-Hopfian groups (in particular, the Baumslag–Solitar group $\langle b , t : t ^ { - 1 } b ^ { 2 } t = b ^ { 3 } \rangle$ is non-Hopfian; cf. also Non-Hopf group); decompositions of $1$-relator groups; Stallings' classification of finitely-generated groups with more than one end in terms of amalgamated products and HNN-extensions; and Stallings' characterization of bipolar structures on groups.

HNN-extensions are of central importance in, e.g., the modern version of the Van Kampen theorem (based on topological results in [a8], [a9]); the Bass–Serre theory of groups acting on trees and the theory of graphs of groups (see [a7]); Dunwoody's accessibility theorem [a3]; and JSJ decompositions of groups [a6].

References

[a1] J.L. Britton, "The word problem" Ann. of Math. , 77 (1963) pp. 16–32
[a2] B. Chandler, W. Magnus, "The history of combinatorial group theory: A case study in the history of ideas" , Studies History Math. and Phys. Sci. , 9 , Springer (1982)
[a3] M.J. Dunwoody, "The accessibility of finitely presented groups" Invent. Math. , 81 (1985) pp. 449–457
[a4] G. Higman, B.H. Neumann, H. Neumann, "Embedding theorems for groups" J. London Math. Soc. , 24 (1949) pp. 247–254; II.4, 13
[a5] R. Lyndon, P. Schupp, "Combinatorial group theory" , Springer (1977)
[a6] E. Rips, Z. Sela, "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition" Ann. of Math. (2) , 146 : 1 (1997) pp. 53–109
[a7] J.P. Serre, "Arbres, amalgams, $S L _ { 2 }$" Astéerisque , 46 (1977)
[a8] E.R. Van Kampen, "On the connection between the fundamental groups of some related spaces" Amer. J. Math. , 55 (1933) pp. 261–267
[a9] E.R. Van Kampen, "On some lemmas in the theory of groups" Amer. J. Math. , 55 (1933) pp. 268–273
How to Cite This Entry:
HNN-extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=HNN-extension&oldid=50277
This article was adapted from an original article by Mike Mihalik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article