On the doubled tetrus

This paper describes some properties of a certain fibered hyperbolic 3manifold which arises ”in nature”; that is, as the cover of a closed manifold constructed by other means. Among other things, the main theorem answers in the affirmative the question of whether it is possible for a fibered hyperbolic manifold to be a double across a connected surface. (We first heard this question from Alan Reid.) Non-hyperbolic fibered doubles are easily constructed by doubling the exterior of a fibered knot across the boundary torus, but producing a fibered hyperbolic double is more subtle problem. In such a manifold the doubling surface — necessarily with genus at least 2 — does not itself admit a fibering and must thus have points of tangency with any fibering, which in particular cannot be invariant under the doubling involution. Our objects of study come from a family of hyperbolic 3-manifolds Mn,k with totally geodesic boundary defined by Paoluzzi-Zimmermann [12]. A well known member of this family is M3,1, which is Thurston’s “tripus” ([15], Ch. 3). Here we consider the ”tetrus” M4,1 (thanks to Richard Kent for naming suggestions). For a manifold M with boundary, let DM denote its double;