### 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 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.

### License

BSD License

### Topics

- Mathematics/Algebra
- Mathematics/Geometry

### Session Buildings

- Prelim
- Algebra
- Simplicial
- Chamber
- Coxeter
- Building

## Cite

×

Buildings-AFP

    @article{Buildings-AFP,
      author  = {Jeremy Sylvestre},
      title   = {Chamber Complexes, Coxeter Systems, and Buildings},
      journal = {Archive of Formal Proofs},
      month   = {July},
      year    = {2016},
      note    = {\url{https://isa-afp.org/entries/Buildings.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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