Handle additions producing essential surfaces

We construct a small, hyperbolic 3-manifold M with the property that, for any integer g ≥ 2, there are infinitely many separating slopes r in ∂M such that the 3-manifold M(r) obtained by attaching a 2-handle to M along r contains an essential separating closed surface of genus g. The resulting manifolds M(r) are still hyperbolic. This contrasts sharply with known finiteness results on Dehn filling and with the known finiteness result on handle addition for the cases g = 0, 1. Our 3-manifold M is the complement of a hyperbolic, small knot in a handlebody of genus 3.