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We calculate the dispersion of a hole in ${\rm CuO_2}$ bilayers and 3Dantiferromagnets, using the self-consistent Born approximation (SCBA) to the$t-J$ model. Superconductivity and the symmetry of its order parameter arestudied introducing a nearest-neighbor density-density attraction induced byshort range antiferromagnetic (AF) fluctuations, as described in recent studiesfor the single layer cuprates. The well-known pairing in the ${ d_{x^2-y^2}}$channel observed for one plane remains robust when three dimensionalinteractions are turned on. In bilayers, as the exchange along the directionperpendicular to the planes grows, eventually a transition to a ``s-wave''state is observed with an order parameter mixture of ${ d_{3z^2-r^2}}$ and ${s_{x^2+y^2 + z^2}}$. For an isotropic 3D antiferromagnet the $d_{x^2 -y^2}$ and$d_{3z^2 - r^2}$ channels are degenerate.Comment: RevTeX (version 3.0), 10 twocolumn pages with 9 embedded figure

physical review. b, condensed matter

Hole dispersion and symmetry of the superconducting order parameter for underdoped<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">CuO</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>bilayers and the three-dimensional antiferromagnets

Hole dispersion and symmetry of the superconducting order parameter for underdopedCuO2bilayers and the three-dimensional antiferromagnets