The Generalized Cuspidal Cohomology Problem (with K.P. Scannell)
(Joint work with Kevin Scannell) It can be shown that the dimension of the space of deformations of a lattice L in SO(3,1) into SO(4,1) is equal to the dimension of cuspidal cohomology with coefficients in R4 (Minkowski space). There are some general results known. For instance, Kapovich observed that if L is a lattice in SO(3,1) generated by two parabolics, then the dimension of cuspidal cohomology with "minkowski coefficients" is 0, and we say that the lattice is rigid. We have studied the Bianchi groups (PSL(2,Od)) and their finite index torsion free subgroups. We can show that cuspidal cohomology is trivial (and hence the lattice is rigid) when d = -1,-2,-3,-7,-11. We have also show via direct computation (using the Mendoza Complex) that the cuspidal cohomology for the figure eight knot complement is trivial, and we have found a link complement whose cuspidal cohomology is non-trivial. Further more, we found a 2 dimensional CW complex in this link complement that will support a deformation. It can be shown that if a manifold contains a totally geodesic that is "not too far from embedded", then bending deformations are possible and hence we will have non-trivial cuspidal cohomology. It then follows that the immersed totally geodesic surfaces in for instance the figure eight knot complement must all be "far from embedded".
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