The virtual 𝑍-representability of certain 3-manifold groups

Baker, Mark D.

doi:10.1090/S0002-9939-1988-0947696-8

The virtual $\textbf {Z}$-representability of certain $3$-manifold groups
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Proc. Amer. Math. Soc. 103 (1988), 996-998 Request permission

Abstract:

We use results on the cohomology of principal congruence subgroups of ${\text {PS}}{{\text {L}}_2}({\mathbf {Z}}[\omega ]),{\omega ^2} + \omega + 1 = 0$, to prove the existence of a large class of closed, orientable $3$-manifolds with virtually ${\mathbf {Z}}$-representable fundamental groups. In particular, these manifolds have finite covers with positive first Betti number.

References

On certain branched cyclic covers of

Fritz Grunewald and Joachim Schwermer, A nonvanishing theorem for the cuspidal cohomology of $\textrm {SL}_{2}$ over imaginary quadratic integers, Math. Ann. 258 (1981/82), no. 2, 183–200. MR 641824, DOI 10.1007/BF01450534
John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619

Coverings of Dehn fillings of surface bundles

Hugh M. Hilden, María Teresa Lozano, and José María Montesinos, On knots that are universal, Topology 24 (1985), no. 4, 499–504. MR 816529, DOI 10.1016/0040-9383(85)90019-9
Robert Riley, A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77 (1975), 281–288. MR 412416, DOI 10.1017/S0305004100051094
Jean-Pierre Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. (2) 92 (1970), 489–527 (French). MR 272790, DOI 10.2307/1970630