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\begin{document}

\title{Chamber complexes, Coxeter systems, and buildings}
\author{Jeremy Sylvestre \\ University of Alberta, Augustana Campus \\ \href{mailto:jeremy.sylvestre@ualberta.ca}{\url{jeremy.sylvestre@ualberta.ca}}}
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\begin{abstract}
We provide a basic formal framework for the theory of chamber complexes and Coxeter systems, and for buildings as thick chamber complexes endowed with a system of apartments. Along the way, we develop some of the general theory of abstract simplicial complexes and of groups (relying on the \textit{group{\_}add} class for the basics), including free groups and group presentations, and their universal properties. The main results verified are that the deletion condition is both necessary and sufficient for a group with a set of generators of order two to be a Coxeter system, and that the apartments in a (thick) building are all uniformly Coxeter.
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\textit{Note:} A number of the proofs in this theory were modelled on or inspired by proofs in the books on buildings by Abramenko and Brown \cite{Abramenko+Brown:Buildings} and by Garrett \cite{Garrett:Buildings}. As well, some of the definitions, statments, and proofs appearing in the first two sections previously appeared in a submission to the \textit{Archive of Formal Proofs} by the author of the current submission \cite{Sylvestre-AFP15}.
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