I will first recall some of the major ideas of Grothendieck-Teichmueller theory as first envisioned by Grothendieck himself. The adjective `geometric' (one could use `nonlinear') refers to the fact that we will consider the full profinite version of the objects involved (in particular the geometric fundamental groups of the moduli stacks of curves), as opposed to the more `linear' or motivic versions afforded by the pronilpotent and prol-l quotients. I will then describe how the profinite topological simplicial complexes appearing in the title enable one to give a rather complete description of certain important groups, prominently the automorphism groups of the profinite Teichmueller modular groups (alias mapping class groups) and their open subgroups.