A Class of Rigid Coxeter Groups

In this paper, we give a new class of rigid Coxeter groups. Let (W, S) be a Coxeter system. Suppose that (0) for each s, t ∈ S such that m(s, t) is even, m(s, t) = 2, (1) for each s 6= t ∈ S such that m(s, t) is odd, {s, t} is a maximal spherical subset of S, (2) there does not exist a three-points subset {s, t, u} ⊂ S such that m(s, t) and m(t, u) are odd, and (3) for each s 6= t ∈ S such that m(s, t) is odd, the number of maximal spherical subsets of S intersecting with {s, t} is at most two, where m(s, t) is the order of st in the Coxeter group W . Then we show that the Coxeter group W is rigid. This is an extension of a result of D. Radcliffe.