  We discuss the superconductor to normal phase transition in an
infinite-layered type-II superconductor in the limit where the Josephson
coupling between layers is negligible. We model each layer as a neutral gas of
thermally excited pancake vortices. We assume the dominant interaction between
vortices in the same and in different layers is the electromagnetic interaction
between the screening currents induced by these vortices. Our main result,
obtained by exactly solving the leading order renormalization group flow, is
that the phase transition in this model is a Kosterlitz--Thouless transition
despite being a three--dimensional system. While the transition itself is
driven by the unbinding of two-dimensional pancake vortices, an RG analysis of
the low temperature phase and a mean-field theory of the high temperature phase
reveal that both phases possess three-dimensional correlations. An experimental
consequence of this is that the jump in the measured in-plane superfluid
stiffness, which is a universal quantity in 2d Kosterlitz-Thouless theory, will
receive a small non--universal correction (of order 1% in
Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$). This overall picture places some claims
expressed in the literature on a more secure analytical footing and also
resolves some conflicting views.
