Higher polyhedral K-groups

We define higher polyhedralK-groups for commutative rings, starting from the stable groups of elementary automorphisms of polyhedral algebras. Both Volodinʹs theory and Quillenʹs + construction are developed. In the special case of algebras associated with unit simplices one recovers the usual algebraic K-groups, while the general case of lattice polytopes reveals many new aspects, governed by polyhedral geometry. This paper is a continuation of Bruns and Gubeladze (Polyhedral K2, Manuscr. Math.) which is devoted to the study of polyhedral aspects of the classical Steinberg relations. The present work explores the polyhedral geometry behind Suslinʹswell known proof of the coincidence of the classical Volodinʹs and Quillenʹs theories. We also determine all K-groups coming from two-dimensional polytopes.