Isometries of hyperbolic Fibonacci manifolds

Abstract

with all subscripts reduced mod m. Some survey of the results about finiteness, arithmeticity, and generalizations of the Fibonacci groups is given in [2]. Application of topological and geometric methods to studying the Fibonacci groups stems from the paper [3] by H. Helling, A. C. Kim and J. Mennicke wherein they constructed a family of three-dimensional manifolds Mn, n >_ 2, such that ~r~(Mn) ~ F(2,2n). Moreover, it was shown in [3] that these manifolds can be equipped with geometric structures of constant curvature. More exactly, the group F(2, 4) ~ Z5 acts by isometries on the spherical space S 3. In this case the manifold M2 is the lens space L(5, 2). The group F(2, 6) is isomorphic to the Euclidean crystallographic group acting on E r In this case the manifold M3 in the Euclidean Hantzsche-Wendt manifold [4]. For n > 4 the group F(2,2n) is isomorphic to a discrete co compact subgroup of PSL(2, C) acting by isometries on the LobachevskiY space I~ 3 without fixed points. For n >_ 4 the manifold Mn is hyperbolic, i.e. it has a metric of constant negative curvature. The manifolds Mn of [3] are ca~ed the Fibonacci manifolds. We recaU some geometric and topological properties of the manifolds Mn. Exact formulae in terms of the Lobachevskil function for the volumes of hyperbolic Fibonacci manifolds were obtained in [5]. It was shown in [6] that for n >_ 2 the manifold Mn is the n-fold cyclic covering of the three-dimensional sphere, branched over the figure-eight knot. Moreover [7], for n >_ 2 the manifold M~ is the two-fold covering of the three-dimensional sphere, branched over the closed three-strings braid (~rx~21) n. In particular, M2 is two-fold branched over the figure-eight knot, and M3 is two-fold branched over the three-component link named Borromean rings. In [8] the Fibonacci manifolds were obtained by 1/0-Dehn surgery on the manifold A4(ff) that is a punctured torus bundle. In this case