Coarse geometry of Coxeter groups

Imagine an observer standing in a rectangular room whose walls, floor, and ceiling are mirrors. It would appear that they are standing in an infinite space, with infinitely many copies of themselves standing at regularly spaced intervals throughout. Geometrically this space is the familiar 3dimensional Euclidean space. Algebraically there are symmetries that shift the point of view between the infinitely many mirror images of the observer. Now imagine that the room is not rectangular. There are still finitely many mirrors surrounding the observer, but intersecting one another in various angles. It will still appear to the observer that they are standing in an infinite space, with infinitely many identical people standing throughout the space in a symmetric pattern, but depending on the angles of the mirrors the space can appear to bend and curve in different ways. The symmetries of spaces that occur this way, also in dimensions other than three, are called reflection groups or Coxeter groups. They occur naturally in various different branches of Mathematics and Physics. This project is about understanding the algebraic properties of Coxeter groups and relating them to the geometric properties of the infinite space that the observer in the mirrored room sees.