Generalized Bowen–Franks groups of integral matrices with the same zeta function

In this paper the relation between the zeta function of an integral matrix and its generalized Bowen–Franks groups is studied. Suppose that A and B are nonnegative integral matrices whose invertible part is diagonalizable over the field of complex numbers and A and B have the same zeta function. Then there is an integer m, which depends only on the zeta function, such that, for any prime q such that gcd(q,m)=1, for any g(x)∈Z[x] with g(0)=1, the q-Sylow subgroup of the generalized Bowen–Franks group BFg(x)(A) and BFg(x)(B) are the same. In particular, if m=1, then zeta function determines generalized Bowen–Franks groups.