The boundedness of the weighted Coxeter group with complete graph

Abstract:

We prove that a weighted Coxeter group $(W,S,L)$ is bounded with $\mathbf {a}(W)=\mathbf {b}’(W):=\max \{L(u),L(w_{s,t})\mid u,s,t\in S,|W_{s,t}|<\infty \}$ if the Coxeter graph of $W$ is complete and $\mathbf {b}’(W)<\infty$, where $W_{s,t}$ is the parabolic subgroup of $W$ generated by $s\ne t$ in $S$ and $w_{s,t}$ is the longest element in $W_{s,t}$ whenever $W_{s,t}$ is finite.