On Auslander–Reiten components and splitting trace lattices for integral group rings

Let G be a finite group and O a complete discrete valuation ring of characteristic zero with residue class field k=O/πO of characteristic p>0. Suppose that O is sufficiently large to satisfy certain conditions and the group ring OG is of infinite representation type. Let Θ be a connected component of the Auslander–Reiten quiver of OG. We show that if Θ contains an OG-lattice M such that M/πM is an indecomposable kG-module and rankOM is not divisible by p, then the tree class of Θ is A∞ and M lies at the end of Θ.