{
\if\help y
  \newlength{\gcolwidth}\setlength{\gcolwidth}{3.8in}
  \newlength{\dcolwidth}\setlength{\dcolwidth}{0.2in}
\else
  \newlength{\gcolwidth}\setlength{\gcolwidth}{3.8in}
  \newlength{\dcolwidth}\setlength{\dcolwidth}{0.2in}
\fi
\newcounter{cl}
\def\class#1#2{%
  \parshape 2 0in \gcolwidth 0.2in \dimexpr\gcolwidth-0.2in\relax \strut%
  \refstepcounter{cl}\label{cl:#1}%
  \raggedright\arabic{cl}.~#2%
}
\newcounter{tf}
\newcommand{\tfmark}[1]{%
  \stepcounter{tf}%
  \customlabel{tf:#1}{\thetf}
  \thetf.
}
\newcommand\tfref[1]{$^\text{\ref{tf:#1}}$}
% rothead following
% http://tex.stackexchange.com/questions/14288/how-combine-make-diagonal-column-heads-in-table-with-multicolumn-headers
\renewcommand{\rothead}[2][60]{\makebox[\dcolwidth][c]{\rotatebox{#1}{\makecell[c]{#2}}}}

% \ding from pifont:
\def\D{\textcolor{blue}{$\pmb\sim$}}
\def\N{\textcolor{red}{\ding{56}}}
\def\U{\textcolor{blue}{?}}
\def\Y{\textcolor{ForestGreen}{\ding{52}}}

\if\help y \begin{table*}[h] \else \begin{table*} \fi
{\begin{center}\small\begin{tabular}{p{2.2in}p{\dcolwidth}p{\dcolwidth}p{\dcolwidth}p{1.6in}}
\parbox[b]{\gcolwidth}{
  \Y$\coloneqq$Yes, \N$\coloneqq$No, \D$\coloneqq$it Depends
  \newline \U$\coloneqq$Unknown (to the author).
  \newline Superscripts: see ``table footnotes'' below.
  \vspace{4mm}
  \newline {\bf Group(s) $G$}
}
  & \rothead{\bf Faithful $Z$?}
  & \rothead{\bf Taylor $Z$?}
  & \rothead{\bf Quadratic?}
  & {\bf See} \\
\hline
\class{torsion}{Finite / torsion groups} &
  \N\tfref{torsion:F} & \Y\tfref{empty} & \Y\tfref{empty} &
  Sec.~\ref{ssec:Trivialities}\\
\class{Zn}{Free Abelian groups $\bbZ^n$} &
  \Y & \Y & \Y & Sec.~\ref{sssec:AbelianGroups} \\
\class{FG}{Free groups $\FG_n$} &
  \Y & \Y & \Y & Sec.~\ref{sssec:FreeGroups} \\
\class{LOF}{LOT and LOF groups} &
  \D & \Y & \Y & Sec.~\ref{ssec:LOF} \\
\class{puretangles}{Knot and pure tangle groups} &
  \D & \Y & \Y & Sec.~\ref{ssec:puretangles} \\
\class{linkgroups}{Link groups} &
  \N\tfref{links:F} & \Y & \Y & Sec.~\ref{ssec:links} \\
\class{2knotgroups}{2-Knots groups} &
  \D & \Y & \Y \\
\class{PB}{Pure braid groups $\PB_n$} &
  \Y & \Y & \Y & Sec.~\ref{ssec:PB} \\
\class{HypAr}{Hyperplane arrangement groups} &
  \U & \Y & \Y & \\
\class{RF}{Reduced free groups $RF_n$} &
  \Y & \Y & \N & Sec.~\ref{ssec:RF} \\
\class{RPB}{Reduced (homotopy) pure  braid groups $\RPB_n$} &
  \Y & \Y & \N \\
\class{PvB}{Pure v-braid groups $\PvB_n$} &
  \U & \N & \Y \\
\class{PwB}{Pure w-braid groups $\PwB_n$} &
  \Y & \Y & \Y & \\
\class{PfB}{Pure f-braid groups $\PfB_n$} &
  & & & Merkov \\
\class{Annular}{Annular braids} &&&& \\
\class{EllPB}{Elliptic pure braid groups $\PB^1_n$ (braids on the torus)} &
  \U & \Y & \N \\
\class{gPB}{Higher genus pure braid groups $\PB^{>1}_n$ (braids on high
  genus surfaces)} &
  \U & \U & \N &
  \arXiv{math/0309245}? \\
\class{CPuB}{Braid commutators $[\PuB_n,\PuB_n]$} &&&&\\
\class{CPvB}{v-Braid commutators $[\PvB_n,\PvB_n]$} &&&&\\
\class{CPwB}{w-Braid commutators $[\PwB_n,\PwB_n]$} &&&&\\
\class{Hilden}{Hilden braids} &&&&\\
\class{MexicanPlait}{Mexican plait braids} &&&& Kurpita-Murasugi \\
\class{Cactus}{Cactus groups} &&&&\\
\class{FundSigma}{Fundamental groups of surfaces} &
  & \Y & \Y \\
\class{MCG}{Mapping class groups} &&&&\\
\class{Torelli}{Torelli groups} &&&& Hain \\
\class{RAAG}{Right-angled Artin groups} & & \Y & \Y \\
\class{Artin}{General Artin groups} & & & \\
\class{BEER}{Groups from BEER} &&&& \arXiv{math/0509661} \\
\class{Brochier}{Groups from Brochier} &&&& \arXiv{1209.0417} \\
\class{PolyFree}{Poly-free groups} &&&& \arXiv{math/0603470} \\
\end{tabular}\end{center}}

\if\help y
\else
{\red MORE: Make sure that all statements are referenced. Additional
columns: $H^2=0$?, extensibly Taylor?. Hierarchical structure for group
list? Add tangles / homology cylinders / w-tangles etc., modulo $C_n$,
$Y_n$, etc.  Add tangles mod concordance, homology cylinders mod homology
cobordism.}
\fi

\caption{Some groups and their expansion properties.} \label{tab:Summary}
{\footnotesize\rm {\bf Table footnotes.}
  \tfmark{torsion:F} Except $G=\{e\}$.
  \tfmark{empty} In an empty manner.
  \tfmark{links:F} Except $G=\bbZ^n$.
}
\end{table*}
}