The twist conjecture for Coxeter groups without small triangle subgroups

Abstract

We prove Mühlherr's twist conjecture for Coxeter systems $(W,S)$ which have no rank $3$ subsystems of type $2$-$3$-$n$ or $2$-$4$-$n$ ($n \geq 3$). In combination with known results this finishes the solution of the isomorphism problem for this class of groups. The condition on the diagram does not allow spherical rank $3$ subsystems, but our result covers “most” of the even Coxeter systems. With respect to earlier contributions, we develop a geometric technique to handle rank $2$ twists, in particular rotation twists which occur in the even case.