  In this paper, we introduce a fast Fourier-Galerkin method for solving
boundary integral equations on torus-shaped surfaces, which are diffeomorphic
to a torus. We analyze the properties of the integral operator's kernel to
derive the decay pattern of the entries in the representation matrix.
Leveraging this decay pattern, we devise a truncation strategy that efficiently
compresses the dense representation matrix of the integral operator into a
sparser form containing only $\mathcal{O}(N\ln^2 N)$ nonzero entries, where $N$
denotes the degrees of freedom of the discretization method. We prove that this
truncation strategy achieves a quasi-optimal convergence order of
$\mathcal{O}(N^{-p/2}\ln N)$, with $p$ representing the degree of regularity of
the exact solution to the boundary integral equation. Additionally, we confirm
that the truncation strategy preserves stability throughout the solution
process. Numerical experiments validate our theoretical findings and
demonstrate the effectiveness of the proposed method.
