Inner Automorphisms of Right-angled Coxeter Groups

If W is a right-angled Coxeter group, then the group Aut(W ) of automorphisms of W acts on the set of conjugacy classes of involutions in W. Following Tits [16], the kernel of this action is denoted by Aut◦(W ). Since W is a CAT(0) group [12], the index of Aut◦(W ) in Aut(W ) is finite and there is a series 1 Inn(W ) Aut◦(W ) Aut(W ) of normal subgroups of Aut(W ). A presentation for Aut◦(W ) was given by Mühlherr in [13]. Our approach is to consider a generating set A obtained as a slight modification of Mühlherr’s. We introduce an effective algorithm for deciding whether a word in the free group F (A) represents an inner automorphism of W under the natural mapping F (A) −→ Aut◦(W ). In other words, we prove the following. Theorem The generalized word problem for Inn(W ) in Aut◦(W ) is solvable.