\section{Some Harder Facts about Power Series and Expansions}
\label{sec:harder}

MORE: This section needs a detailed look.


The lower central series $G_n$ of $G$ is defined inductively by
setting $G_1\coloneqq G$ and $G_{n+1}\coloneqq (G,G_n)=\{(x,y)\colon
x\in G,\, y\in G_n\}$. It is clear that $G=G_1\rhd G_2\rhd
G_3\rhd \ldots$, and that the quotients $G_n/G_{n+1}$ are Abelian
groups. It is well known that the group commutator $(x,y)$ induces
a structure of a graded Lie ring on $\calL G\coloneqq\bigoplus_n
G_n/G_{n+1}$ (see e.g.~\cite{MagnusKarrassSolitar:CGT}).
Proposition~\ref{prop:StrongerCommutators} implies that the map
$x\mapsto\bar{x}$ maps $G_n$ to $I^n$ and induces a Lie morphism $\calL
G\to\calA(G)$ and hence an algebra morphism $\calU(\bbQ\otimes_\bbZ\calL
G)\to\calA(G)$, where $\calU$ denotes the universal enveloping algebra.
Quillen~\cite{Quillen:OnGrOfAGroupRing} proves that that morphism is in
fact an isomorphism: $\calU(\bbQ\otimes_\bbZ\calL G)\cong\calA(G)$.

Note that if $F=F(x_i)$ is a free group on some set of generators $(x_i)$
and $F_n$ denotes the lower central series of $F$, then $F_1/F_2$ is the
free Abelian group with generators $x_i$ and $F_2/F_3$ is the free Abelian
group with generators $(x_i,x_j)$ for $i<j$ (or allow $(x_i,x_j)$ with
arbitrary $i,j$, yet note that modulo $F_3$ and using Abelian notation,
$(x_i,x_j)+(x_j,x_i)=0$). We let $V=V(x_i)$ be the $\bbQ$-vector space with
basis $(x_i)$, and note that $\bbQ\otimes(F_1/F_2)\cong V$ and
$\bbQ\otimes(F_2/F_3)\cong \bigwedge^2 V\subset V\otimes V$.

\begin{definition} We say that a presentation $G=\langle x_i\mid
r_k\rangle$ of a group $G$ is ``quadratically efficient'' if the relations
$r_k$, in themselves elements of the free group $F=F(x_i)$, all belong
to the commutator subgroup $F_2=(F,F)$ of $F$ and their images $\rho_k$
in $\bbQ\otimes(F_2/F_3)=\bigwedge^2V$ are linearly independent.
\end{definition}

\begin{theorem} If a group $G$ has a quadratically efficient presentation
$\langle x_i\mid r_k\rangle$ then it is quadratic and $\calA(G)\cong
TV/\langle\rho_k\rangle$ is the tensor algebra $TV=\bigoplus_n V^{\otimes
n}$ of $V=V(x_i)$ modulo the ideal generated by the images $\rho_k$
of the relations $r_k$ in $\bigwedge^2V\subset V^{\otimes 2}$.
\end{theorem}

MORE: Examples and proof.

MORE Quillen's theorem (using expansions?). Must sort in Quillen's theorem
and link up with existing literature, expecially with Suciu-Wang.

MORE Hutchings-Positselski-Lee.