Some new biautomatic Coxeter groups

A Coxeter system is called two-dimensional if every parabolic subgroup of rank at least 3 is infinite. We apply results of Niblo and Reeves and of B.T. Williams to show that Coxeter groups with two-dimensional systems are biautomatic if the corresponding presentation contains no visible Euclidean “triangle” groups. This provides a partial positive answer to a question posed by Niblo and Reeves in [G. Niblo, L. Reeves, Coxeter groups act on CAT(0) cube complexes, J. Group Theory 6 (3) (2003) 399–413].