Discrete automorphism groups of convex cones of finite type

TL;DR

This paper studies subgroups of SL(n,Z) that preserve convex cones, focusing on their geometric and algebraic properties, with applications to arithmetic groups, Kac-Moody algebras, and algebraic geometry.

Contribution

It characterizes automorphism groups of convex cones of finite type, highlighting their structure and examples in various mathematical contexts.

Findings

Identifies conditions for subgroups to preserve convex cones

Provides examples from arithmetic groups and algebraic geometry

Connects automorphism groups with Kac-Moody algebra Weyl groups

Abstract

We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras and do occur in algebraic geometry as the automorphism groups of projective manifolds acting on their ample cones.