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The role of generalized Bowen-Franks groups (BF-groups) as topologicalconjugacy invariants for $\mathbb{T}^{n}$ automorphisms is studied. Usingalgebraic number theory, a link is established between equality of BF-groupsfor different automorphisms ($BF$-equivalence) and an identical position in afinite lattice ($\mathcal{L}$-equivalence). Important cases of equivalence ofthe two conditions are proved. Finally, a topological interpretation of theclassical BF-group $\mathbb{Z}% ^{n}/\mathbb{Z}^{n}(I-A)$ in this context ispresented.

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