On hyperelliptic involutions of hyperbolic 3-manifolds

Abstract

By classical results, a compact Riemann surface, or equivalently an orientable closed hyperbolic 2-manifold, has at most one hyperelliptic involution (i.e. the quotient by the involution is the 2-sphere). A hyperelliptic involution is central in the automorphism or isometry group of such a hyperelliptic surface, its fixed points are exactly the Weierstrass points of the surface (see e.g. [FK]). In the present paper, we consider hyperelliptic involutions of hyperbolic 3manifolds; by definition, these are isometric involutions whose quotient is the 3-sphere. The situation is more complicated in dimension three, in particular a hyperbolic 3-manifold can have more than one hyperelliptic involution. Our main result states that there is a universal bound on the number of non-conjugate hyperelliptic involutions of closed hyperbolic 3-manifolds; this bound is quite small, and we compute it for various situations. Our methods also determine how different hyperelliptic involutions are related. There remains a small gap between these upper bounds and the number of different involutionswhichwe can actually construct (see [R1],[RZ]).We remark that the question on the number of different hyperelliptic involutions of hyperbolic 3-manifolds has explicitly been asked in [MV]. We obtain also results for hyperelliptic isometries of orders greater than two; in particular, for prime numbers p > 2, we give a complete solution to the problem on the number of different hyperelliptic isometries of order p (i.e. the upper bound we obtain is sharp). Let Isom+(M) denote the orientation preserving isometry group of a hyperbolic 3-manifoldM .