Skip to main content\(\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\Inn}{Inn}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Perm}{Perm}
\DeclareMathOperator{\Stab}{Stab}
\DeclareMathOperator{\Orb}{Orb}
\DeclareMathOperator{\Rot}{Rot}
\DeclareMathOperator{\re}{Re}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\img}{image}
\DeclareMathOperator{\conj}{conj}
\DeclareMathOperator{\Id}{Id}
\def\expi{E}
\def\wrap{W}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Quat}{\mathbb{H}}
\newcommand{\extC}{\hat{\C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\extR}{\hat{\R}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\extF}{\hat{\F}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Proj}{\mathbb{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\M}{{\rm MOB}}
\newcommand{\E}{{\rm EUC}}
\renewcommand{\H}{{\rm HYP}}
\newcommand{\HU}{\rm HYP_{\U}}
\renewcommand{\S}{{\rm ELL}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\closedD}{\hat{\D}}
\newcommand{\U}{\mathbb{U}}
\newcommand{\spacer}{\rule[0cm]{0cm}{0cm}}
\newcommand{\MOD}{\mathbin{\text{MOD}}}
\newcommand{\twotwo}[4]{\left[ \begin{array}{cc} #1 \amp #2 \\
#3 \amp #4 \end{array} \right]}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
