Reflection of a Coxeter group

A reflection of a Coxeter system $(W,S)$ is a conjugate $ws{w}^{-1}$ of a simple generator $s\in S$.

The wall associated with a reflection $r$ consists of the Cayley-graph edges fixed setwise and reversed by left multiplication by $r$. Its complement has two connected components.

The two connected components of the complement of a wall of a Coxeter group are its half-spaces. A wall separates two group elements when they lie in opposite half-spaces.