Hyperbolic groups containing subgroups of type $\mathscr{F}_{3}$ not $\mathscr{F}_{4}$

Abstract

We give examples of hyperbolic groups which contain subgroups that are of type F 3 but not of type F 4 . These groups are obtained by Dehn filling starting from a non-uniform lattice in PO(8, 1) which was previously studied by Italiano, Martelli and Migliorini.

Algebraic fibrings of a hyperbolic $7$-manifold

“…We also remark again that Llosa Isenrich, Martelli, and Py construct an algebraic fibring of an 8-dimensional hyperbolic manifold with kernel of type F 3 but not F 4 in [IMP21]. The manifold here is obtained by Dehn filling M 8 .…”

“…Remark. In [IMP21], Llosa Isenrich, Martelli, and Py construct a hyperbolic 8-manifold M that algebraically fibres with kernel of type F 3 but not F 4 . The manifold M is obtained by Dehn filling M 8 and provides the first example of a hyperbolic group with a subgroup of type F 3 but not F 4 .…”

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).…”

“…These are particular cases of the construction in [10], and thus 𝐺 is here the fundamental group of a negatively curved pseudo-manifold of dimension 8. Note that in order to obtain infinitely many examples in [10, Theorem 1], a delicate argument about perturbing the fibration was required.…”

“…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.…”