On a hyperbolic 3-manifold with some special properties

Source abstract

We present a closed hyperbolic 3-manifold M with some surprising properties. The universal covering group of M is a normal torsion-free subgroup of minimal index in one of the nine Coxeter groups G , generated by the reflections in the faces of one of the nine Lannér-tetrahedra (bounded tetrahedra in hyperbolic 3-space all of whose dihedral angles are of the form π/ n with n ∈ ℕ see [1] or [3]). The corresponding Coxeter group G splits as a semidirect product G = π 1 M ⋉ A , where A is a finite subgroup of G , and G is the only one of the nine Coxeter groups associated to the Lannér-tetrahedra which admits such a splitting (this follows using results in [4]). We derive a presentation of π 1 M and show that the first homology group H 1 ( M ) of M is isomorphic to ℚ 11 . This is in sharp contrast to other torsion-free (non-normal) subgroups of finite index in Coxeter groups constructed in [1] which all have finite first homology (though it is known that they are all virtually ℚ-representable (see [5], p. 434). It follows from our computations that the Heegaard genus of M is 11, and that there exists a Heegaard splitting of M of genus 11 invariant under the action of the group I + ( M ) ≌ S 5 ⊕ ℚ 2 of orientation-preserving isometries of M (we compute this group in [4]), so that the Heegaard genus of M is equal to the equivariant Heegaard genus of the action of I + ( M ) on M . Moreover M is maximally symmetric in the sense of [4, 6]: the order 120 of the subgroup of index 2 in I + ( M ) which preserves both handle-bodies of the Heegaard splitting is the maximal possible order of a group of orientation-preserving diffeomorphisms of a handle-body of genus 11. (This maximal order is 12( g—1 ) for a handle-body of genus g ; see [7].) By taking the coverings M q of M corresponding to the surjections π 1 M→H 1 ( M ) ≌ ℚ 11 →(ℚ q ) 11 for q ∈ ℕ, we obtain explicitly an infinite series of maximally symmetric hyperbolic 3-manifolds.