Abstract :
In [Vi4] Vinberg has given a lemma relating image-lattices of signature (n, 1) having a cofinite reflection group with certain positive definite, reflective sublattices, i.e. sublattices having a root system of maximal rank. This paper contains a detailed investigation of the existence of non reflective positive definite lattices in high dimensional genera. As a consequence of these results and Vinbergʹs lemma, the maximal dimension of image-lattices of signature (n, 1) with a cofinite reflection group is determined to ben+1=22.
