We construct a hyperbolic 3-manifold M (with partial derivative M totally geodesic) which contains no essential closed surfaces, but for any even integer g > 0, there are infinitely many separating slopes r on partial derivative M so that M[r], the 3-manifold obtained by attaching 2-handle to M along r, contains an essential separating closed surface of genus g and is still hyperbolic. The result contrasts sharply with those known finiteness results for the cases g = 0, 1. Our 3-manifold M is the complement of a simple small knot in a handlebody.
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