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\begin{document}


\section*{Lecture 2}

\subsubsection*{Stronger NFT}

Choose right coset representatives for $C\setminus A$, $Ca_1,Ca_2,\ldots\, $
and $C\setminus B$, $Cb_1, Cb_2,\ldots\,$ This gives a unique normal form
$a_0b_1a_2b_3\cdots\;$ or $b_0a_1b_2a_3\cdots\;$ with the property that
each $a_i,b_i$ is one of the coset representatives.

{\sc Remark. } This is not good either. It rather should be
$ca_0b_1a_2b_3\cdots\;$

Existence of ``ordinary'' normal forms: each
$g\in A\, \substack{ * \\ C }\, B$
equals $g_1g_2\cdots g_k$, $g_i\in A\sqcup B$. Continually ``combine
like terms'' ($b_1a_2a_3b_4\longrightarrow b_1(a_2a_3)b_4$) and combine
elements of $C$ with neighboring elements
($a_1\Phi_b(c)a_2\to a_1\Phi_a(c)a_2\to (a_1\Phi_a(c))a_2$).

Existence of ``stronger'' normal forms: given an ``ordinary'' normal form
$g_0g_1\cdots g_3g_4$ we continually replace $g$'s with suitable coset
representatives, starting with $g_4$:
$$
g_0g_1\cdots g_3g_4\xrightarrow{g_4\in Ca_4}g_0g_1\cdots (g_3c_4)a_4
\xrightarrow{g_3c_4\in Cb_3}g_0g_1\cdots c_3b_3a_4\to\ldots\to
ca_0b_1\cdots b_3a_4
$$

\subsection*{HNN Extensions}

The idea is to construct something given
$$
{\xymatrix{
A&C\ar@/_/@{->}[l]_{\Phi_1}\ar@/^/@{->}[l]^{\Phi_2}\\
}}
\qquad
\mathrm{or}
\qquad
{\xymatrix{
C_1 \ar@{^{(}->}[dr] \ar@{->}[rr]^{\Phi}&&C_2\ar@{^{(}->}[dl]\\
&A&\\
}}
$$
\begin{definition}
The HNN extension (HNN stands for Higman--Neumann-Neumann) associated
to this data is the group $A*_C$ or $A*_{\Phi}$
$$
\langle A,t|c^t=\Phi(c)\,\forall c\in C_1\rangle
\cong
\left.
\begin{matrix}
&A*\langle t\rangle\\
&
\end{matrix}
\right/
\begin{matrix}
&\\
\langle\langle c^t\Phi(c^{-1}),c\in C_1\rangle\rangle.&
\end{matrix}
$$
$t$ is called stable letter.
\end{definition}

\medskip
\subsubsection*{NFT for HNN Extensions}
\begin{theorem}
Every element in $A*_C$ can be expressed as
$a_0t^{\varepsilon_1}a_1t^{\varepsilon_2}a_2t^{\varepsilon_3}\cdots$,
each $\varepsilon_i=\pm 1$, each $a_i\in A$, and there is no
appearance of\; $t^{-1}c_1t,\, c_1\in C_1$ and\; $tc_2t^{-1},\, c_2\in C_2$.
\end{theorem}
Length of $g$ is number of $t$'s in its normal form. Length is invariant.

A imbeds in $A*_C$.

{\sc Example. } Let $A=A_5$, $\beta=(12)(34),\,\gamma=(123)$. Then
$\langle A,t|\beta^t=\gamma\rangle\cong \mathbb Z$.

\subsection*{Hopfian groups.}
\begin{definition}
A group $G$ is hopfian if every epimorphism $G\to G$ is a monomorphism.
\end{definition}
{\sc Exercise. } Finitely generated abelian group is hopfian.

{\sc Exercise. } $\mathbb Z^\infty, F_\infty$ are not hopfian. (Hint: kill
one generator.)
\begin{definition}
$\bs(m,n)=\langle a,t|(a^m)^t=a^n\rangle$ ($m,n\neq 0$). (BS stands for
Baumslag--Solitar.)
$$
{\xymatrix{
m\mathbb Z \ar@{^{(}->}[dr] \ar@{->}[rr]^{\Phi=\cdot\frac{n}{m}}&&n\mathbb Z\ar@{^{(}->}[dl]\\
&\mathbb Z&\\
}}
$$
\end{definition}
$G=\bs(2,3)$ is not hopfian. Consider homomorphism $\psi :\,G\to G$ induced by
$t\mapsto t,\,a\mapsto a^2$. Then the relation
$[(a^2)^t=a^3]\mapsto [(a^4)^t=a^3]$, so $\psi$ is really a homomorphism.
Further, $\psi$ is surjective since $t=\psi(t),\;a=\psi(a^{-1}(a)^t)$, but not
injective, since $1\neq[a^t,a]$ maps to $1$. Show that.\newline
$[a^t,a]=t^{-1}a^{-1}ta^{-1}t^{-1}ata$ --- is a normal form of length $>0$.\newline
$[a^t,a]\mapsto [(a^2)^t,a^2]=[a^3,a^2]=1$.

{\sc Reminder. } $x^y=y^{-1}xy$,\;$[x,y]=x^{-1}y^{-1}xy$.

{\sc Exercise. } $\bs(1,n)$ is hopfian, $\bs(n,n)$ is hopfian.

Constructions of amalgamated product and HNN extension are believed to
be ``equally strong''. In the following example we present a group both
as an amalgamated product and as a HNN extension.

{\sc Example. } $G=\langle a,b,c,d|aba^{-1}b^{-1}cdc^{-1}d^{-1}\rangle$
\begin{verse}
$\bullet$\; $G\cong F_2\, \substack{ * \\ \mathbb Z }\, F_2\cong
\langle a,b\rangle \,
\substack{ * \\ \mathbb Z }\,\langle c,d\rangle$,\qquad
($\mathbb Z=\langle [a,b]\rangle= \langle [c,d]\rangle$).\\
$\bullet$\; $G\cong F_3*_\mathbb Z$, given by
%$\langle b\rangle\subseteq
%\langle b,c,d\rangle,\;
%\langle b^{-1}cdc^{-1}d^{-1}\rangle\subseteq
%\langle b,c,d\rangle$,\;
%$\varphi:\langle b^{-1}cdc^{-1}d^{-1}\rangle\to \langle b\rangle$,\;
%$\varphi:b^{-1}cdc^{-1}d^{-1}\mapsto  b^{-1}$,\;
%$(b^{-1}cdc^{-1}d^{-1})^a=b^{-1}$.
$$
{\xymatrix{
\langle b^{-1}cdc^{-1}d^{-1}\rangle \ar@{^{(}->}[dr]
\ar@{->}[rr]^{\phantom{blah}\varphi:b^{-1}cdc^{-1}d^{-1}\mapsto  b^{-1} }&&
\langle b\rangle\ar@{^{(}->}[dl]\\
&\langle b,c,d\rangle&\\
}}
$$
and stable letter $a$, that is $(b^{-1}cdc^{-1}d^{-1})^a=b^{-1}$.
\end{verse}

\subsection*{A Graph of Groups}
Let $\Gamma$ be a directed graph. For each vertex $v\in \Gamma$, we have a
group $G_v$. For each edge $e\in \Gamma$, we have  group $G_e$. An edge is
attached at its initial vertex $i(e)$ and terminal vertex $t(e)$.

$G_e$ has monomorphisms to $G_{i(e)}, G_{t(e)}$.\\
$$\varphi_{e+}:G_e\to G_{t(e)},\:
\varphi_{e-}:G_e\to G_{i(e)}$$

From this data you can get a group $\pi_1\Gamma$,
$\pi_1(\mathrm{graph\,of\,groups})$.

Consider all combinatorial paths starting at some fixed vertex of the graph
(see figure): $abc,abe^{-1}d^{-1},\ldots$

Equivalence relation is generated by removal of backtracks:
$abcc^{-1}e^{-1}\sim abe^{-1}$. So in our example, we get a subgroup
$\langle a,b,c,d,e|-\rangle$ generated by
$\langle abc, dec^{-1}\rangle$.

Label each edge a ``stable'' letter $t_i$, pick a base point.

Our group is the set of equivalence classes of ``words'' of the form
$h_0s_1h_1s_2h_2\cdots h_ks_kh_{k+1}$, where each
$s_i=\mathrm{some\,}t_j^{\pm1}$.

If $ht_ih'$ is in the word, then
$h\in G_{i(e_i)},\,h'\in G_{t(e_i)}$;\par
if $h't_i^{-1}h$ is in the word, then
$h\in G_{i(e_i)},\,h'\in G_{t(e_i)}$.

Equivalence relation is generated by ``backtracks'':
$$\varphi_{i-}(g_i)t_i\sim t_i\varphi_{i+}(g_i)\qquad \forall g_i\in G_i=G_{e_i}.$$
\end{document}
