  In this paper we introduce the galaxy of Coxeter groups -- an infinite
dimensional, locally finite, ranked simplicial complex which captures
isomorphisms between Coxeter systems. In doing so, we would like to suggest a
new framework to study the isomorphism problem for Coxeter groups. We prove
some structural results about this space, provide a full characterization in
small ranks and propose many questions. In addition we survey known tools,
results and conjectures. Along the way we show profinite rigidity of triangle
Coxeter groups -- a result which is possibly of independent interest.
