Lifted heegaard surfaces and virtually haken manifolds

摘要

In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements [A. Casson and C. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275283] and extends [J. Masters, W. Menasco and X. Zhang, Heegaard splittings and virtually Haken Dehn filling, New York J. Math. 10 (2004) 133150].