\section{Some Specific Families of Groups}
\label{sec:specific}

{\red
MORE: Sort in: homologically trivial braids in the torus / in genus $g$,
upper McCool.
}

\subsection{LOT and LOF groups} \label{ssec:LOF} \ 

\vskip 1mm
\parshape 1 \parindent \quotewidth
{\small\noindent{\bf Summary. }
  Howie~\cite{Howie:RibbonDiscComplements, Howie:HigherRibbonKnots}
  defines a class of groups associated with certain ``labelled oriented
  graphs'' $\Gamma$, and studies in detail the case when $\Gamma$ is a
  tree, calling the resulting class of groups ``LOT groups'', showing
  that they are the fundamental groups of ribbon d-knots, for $d\geq
  2$. We allow forests instead of just trees, call the resulting class
  ``LOF groups'', and study their expansions.
}
\vskip 1mm

\noindent Following Howie~\cite[Section~3]{Howie:RibbonDiscComplements},
a ``labeled oriented graph'' $\Gamma$ is a quintuple
$\Gamma=(V,E,\iota,\tau,\lambda)$, where $V$ and $E$ are
sets of ``vertices'' and ``edges'' respectively (finite,
in~\cite{Howie:RibbonDiscComplements}, but not necessarily so,
for us), where $\iota$ and $\tau$ are maps $E\to V$ which map every
edge $a\in E$ to its initial vertex $\iota(a)$ and terminal vertex
$\tau(a)$\footnote{Having established notation we will use graph
theoretic language with no further comment.}, and where $\lambda\colon
E\to V^{\pm 1} \coloneqq \{a,a^{-1}\colon a\in V\}$ puts an additional
``label'', which is either a vertex or the formal inverse of a vertex,
conventionaly marked near the middle of the edge. To such $\Gamma$
we associate a group $G(\Gamma)$, defined as the group whose set of
generators is $V$ and whose relations correspond to the edges of $\Gamma$,
where the relation for an edge $a$ with $\iota(a)=x$, $\tau(a)=y$,
and $\lambda(a)=z^{\pm 1}$, namely for $x\xrightarrow{z^{\pm 1}}y$,
is $x=z^{\mp 1}yz^{\pm 1}=y^{z^{\pm 1}}$, or ``the tail is the head
conjugated by the middle''\footnotemark. Here is a simple example of a
lalebled graph with two connected components, and the corresponding group
presentation:

\[
  \def\gens{$x_1,x_2,y_1,y_2,y_3,y_4,z_1,z_2$}
  \def\relsA{$r_1\colon y_1=x_1^{z_2^{-1}},\, r_2\colon y_2=x_2^{x_1}$}
  \def\relsB{$r_3\colon y_3=x_2^{z_1},\  r_4\colon y_4=x_2^{z_2}$}
  \def\relsC{$r_5\colon z_1=y_4^{y_1},\, r_6\colon z_2=y_4^{z_1^{-1}}$}
  \input{figs/LOFExample.pdf_t}
\]

\footnotetext{Note that the relation corresponding to
$x\xrightarrow{z^{-1}}y$ is equivalent to the relation
for $x\xleftarrow{z}y$, so we could have restricted, as
Howie~\cite{Howie:RibbonDiscComplements} does, to middle labels with
positive powers, at the cost of reversing some edge orientations.
}

We say that $\Gamma$ is a tree if its underlying graph $(V,E,\iota.\tau)$
is a tree. In this case, Howie~\cite{Howie:RibbonDiscComplements,
Howie:HigherRibbonKnots} calls $G(\Gamma)$ a LOT (Labelled Oriented Tree)
group. Howie shows that such groups are precisely the fundamental groups
of ribbon $d$-knots in $S^{d+2}$, for $d\geq 2$.

We say that $\Gamma$ is a forest of rank $n$ if its underlying
graph $(V,E,\iota.\tau)$ is a disjoint union of $n$ trees, and call
the corresponding groups ``LOF groups'' of rank $n$ (note that the
labeling $\lambda$ can jump across components). One may show (see
also~\cite[Comment~3.10]{KBH}) that such groups are precisely the
fundamental groups of ribbon knottings of wedge sums of $n$ based
$d$-spheres in $S^{d+2}$, for $d\geq 2$.

For any $\Gamma$, the edge relations in $G\coloneqq G(\Gamma)$ imply that
it is normally generated by one generator for each connected component of
$\Gamma$, and hence by Corollary~\ref{cor:gens} $\calA \coloneqq \calA(G)$
is generated by one generator for each component of $\Gamma$. If $\Gamma$
is not a forest, that's all that we can say at this point. If $\Gamma$
is a forest, let $\{x_i\}_{i=1}^n\subset V$ be some choice of roots
for the components of $\Gamma$. Then $\calA$ is generated by the
elements $\barx_i=[x_i-1]$ in $\calA_1=I/I^2$, and we guess that the
$\barx_i$'s freely generate $\calA$. To verify this we set $A \coloneqq
\FA(\barx_i)$, the free associative algebra generated by the $\barx_i$'s,
note the obvious projection $\pi\colon A\to\calA$, and construct (below)
an $A$-expansion $Z_A\colon G\to A$ (see Section~\ref{ssec:AExpansions}).

We construct $Z_A$ degree by degree, in the spirit of
Section~\ref{sec:dbd} (though without using the results of that section).

The beginning of the construction is forced by
Proposition~\ref{prop:Deg1}: we must have $Z_1(g)=1+\bar{g}\in\calA_{\leq
1}(G)$ for every $g\in G$, so we must have $Z_{A,1}(y)=1+\bar{y}\in
A_{\leq 1}$ for every generator $y$ of $G$.  If $y$ is one
of these generators and $x_i$ is the root of the tree that $y$
belongs to, then by the relations, $y$ is conjugate to $x_i$, so by
Proposition~\ref{prop:gh}, $\bar{y}=\barx_i$ in $\calA_{\leq 1}(G)$ and
we must set $Z_{A,1}(y)=1+\barx_i$. So considering the example before
to degree $1$, we must have:
\[ \def\x{1+\barx} \input{figs/LOFDeg1.pdf_t} \]

Now assume that we found and extension $Z_{A,d}$ of $Z_{A,1}$ to degree
$d$; we aim to extend it further to degree $d+1$. Using
Capsule~\ref{cap:GroupLike} find $\phi_i\in A_{d+1}$ so that on the roots
$x_i$ we'd have that $Z_{A,d+1}(x_i)\coloneqq Z_{A,d}(x_i)+\phi_i$ is group
like. So far we have (dropping one connected component to save space):
\[ \def\Z{Z_{A,d}} \input{figs/LOFDegd.pdf_t} \]

But now the values of $Z_{A,d+1}$ on the immediate neighbors of the roots
($(y_2,y_3,y_4)$ in the partial example) are determined: they have to be
conjugates of the values on the roots as specified by the edge relations.
The values of the conjugators $(x_1,z_1,z_2)$ in these edge relations
might already be specified only to degree $d$, but by
Proposition~\ref{prop:conj}, that is enough. Continuing in this way the
values of $Z_{A,d+1}$ on farther and farther neighbors of the roots are
determined, and eventually $Z_{A,d+1}$ is fully determined and is a Taylor
expansion to degree $d+1$.

In summary, we have proven the following:

\begin{theorem} If $G=G(\Gamma)$ is a LOF group and $\{x_i\}$ is a choice
of roots for the components of $\Gamma$, then $\calA(G)$ is a free
associative algebra with generators $\{\barx_i\}$ in bijection with the
roots, and the Taylor expansions for $G$ are in a bijection with choices of
group-like elements $\{Z(x_i)\}$ in $\calA(G)$, one for each root, such
that to degree 1, $Z(x_i)=1+\barx_i$.
\end{theorem}

\begin{remark} As $\calA(G)$ is free, LOF groups are always quadratic.
\end{remark}

\begin{remark} \label{rem:LOFZ} Capsule~\ref{cap:GroupLike} breaks over
$\bbZ$, and indeed in general Taylor expansions for LOF groups do not exist
over $\bbZ$. Otherwise our construction works in an almost verbatim manner
to construct multiplicative expansions for LOF groups over $\bbZ$.
\end{remark}

{\red MORE. Faithfulness (ask Gwenael?)? $n=\infty$?}

\subsection{Knot and Pure Tangle} \label{ssec:puretangles}

\subsection{Link Groups} \label{ssec:links}

{\red MORE: For link groups, state a theorem about the relationship with Milnor invariants;
perhaps prove Stallings' using expansions?}

\subsection{Reduced Free Groups} \label{ssec:RF}

{\red MORE.}