Reflection groups and polytopes over finite fields. I. (Q1883381)

Suppose that \(\Gamma\) is a (possibly infinite) Coxeter group of rank \(n\), with string diagram. By its action on a root lattice, \(\Gamma\) is isomorphic to some \(G\leq\text{GL}_n(\mathbb{Z})\). Reducing modulo a prime \(p\geq 3\), this gives a finite group \(G^p\) generated by \(n\) reflections in \(\text{GL}_n(\mathbb{Z}_p)\). -- If \(G^p\) is the automorphism group of an abstract regular polytope, then it is called a string \(C\)-group. The authors give an overview of abstract regular polytopes and they give conditions under which for a crystallographic Coxeter group \(G\) the group \(G^p\) is a string \(C\)-group. In particular, this is the case if \(n=1,2\), or \(3\).