The spectrum of global representations for families of bounded rank and VI-modules

A global representation is a compatible collection of representations of the outer automorphism groups of the finite groups belonging to a family U. These arise in classical representation theory, in the study of representation stability, as well as in global homotopy theory. In this paper we begin a systematic study of the derived category D (U;k) of global representations over fields k of characteristic zero, from the point-of-view of tensor-triangular geometry. We calculate its Balmer spectrum for various infinite families of finite groups including elementary abelian p-groups, cyclic groups, and finite abelian p-groups of bounded rank. We then deduce that the Balmer spectrum associated to the family of finite abelian p-groups has infinite Krull dimension and infinite Cantor--Bendixson rank, illustrating the complex phenomena we encounter. As a concrete application, we provide a complete tt-theoretic classification of finitely generated derived VI-modules. Our proofs rely on subtle information about the growth behaviour of global representations studied in a companion paper, as well as novel methods from non-rigid tt-geometry.