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In this paper, we give infinitely many non-Haken hyperbolic genus three3-manifolds each of which has a finite cover whose induced Heegaard surfacefrom some genus three Heegaard surface of the base manifold is reducible butcan be compressed into an incompressible surface. This result supplements [CG]and extends [MMZ].

LIFTED HEEGAARD SURFACES AND VIRTUALLY HAKEN MANIFOLDS