The Boundedness Of Some Weighted Coxeter Groups

About the boufndedfness of a weighted Coxeter group,Lusztig proposed a con-jecture:any weighted Coxeter group(W,S,L)admits a bound N(W):=max{L(wI)|I(?)S,|WI|<∞},where |X| is the cardinality of a set X,WI is the parabolic subgroup of W generated by I(?)S,WI is the longest element in WI whenever|WI|<∞.In this paper,we verify the conjecture of Lusztig in two cases:(i)Coxeter graph of(W,S,L)is non-3-edge-labeling(i.e.the order mst of the product st is not 3 for any s,t ∈S);(ii)S={a,b,c,d},mac=2 and 3≤mab,mbc,mcd,mda,mbd≤∞.