Hyperbolic 5-manifolds that fiber over $$S^1$$

Abstract

We exhibit some finite-volume cusped hyperbolic 5-manifolds that fiber over the circle. These include the smallest hyperbolic 5-manifold known, discovered by Ratcliffe and Tschantz. As a consequence, we build a finite type subgroup of a hyperbolic group that is not hyperbolic.

Generalizations of Topological Decomposition and Zeno Sequence in Fibered n-Spaces

“…However, there is restriction on the fibrations over n-manifolds. In case of an n-manifold, if it is n − even (i.e., Minkowski 4-space), then it cannot be fibered over S 1 due to the Chern-Gauss-Bonnet theorem [9]. Furthermore, it is known that even in the case of a Euclidean 4-manifold E 4 , the manifold E 4 cannot be fibered by a 2-dimensional fiber [10].…”

Special IMM groups

“…In this section, we recall some important facts about the manifold examples constructed by Italiano-Martelli-Migliorini in [11] and [12] and by Llosa Isenrich-Martelli-Py in [10]. These examples satisfy the following hypotheses (after possibly passing to a finite sheeted cover).…”

“…The examples in [10] have the property that ker(𝑓) is type  3 but not type  4 . The examples in [12] have the property that ker(𝑓) has type  .…”

“…The hyperbolic manifold examples constructed in [12] and [10] are commensurable with right-angled Coxeter groups, so their virtual special-ness is immediate. However, the manifolds constructed are only finite volume and not compact, so [12] and [10] must apply Dehn filling techniques to obtain hyperbolic groups with the appropriate subgroup structure. By using a version of the Malnormal Special Quotient Theorem due to Wise [17], we will explain how this Dehn filling can be chosen to preserve virtual special-ness.…”

Homology growth, hyperbolization, and fibering

“…There has been some recent progress towards determining whether or not some analogue of this phenomenon persists in higher dimensions. In a geometric direction, Italiano, Martelli, and Migliorini constructed the first finite volume hyperbolic 5-manifolds that fiber over a circle [35], while in a more algebraic direction, Kielak [43] and Fisher [27] established a close connection between homology growth in finite covers and virtual fibering. But, there is a curious 1 dearth of odd dimensional examples that do not virtually fiber, even if one passes from the hyperbolic to the more flexible Gromov hyperbolic setting.…”