Bowen–Franks groups as conjugacy invariants for $\mathbb{T}^{n}$ -automorphisms

Summary.

The role of generalized Bowen–Franks groups (BF-groups) as topological conjugacy invariants for $\mathbb{T}^{n}$ -automorphisms is studied, including a topological interpretation of the classical BF-group $\mathbb{Z}^{n} /\mathbb{Z}^{n} (I - A)$ in this context.

Using algebraic number theory, a link is established between equality of BF-groups for different automorphisms (BF-equivalence) and an identical position in a finite lattice ( $\mathcal{L}$ -equivalence). Important cases of equivalence of the two conditions are proved.