Exotic subgroups of hyperbolic groups

TL;DR

This work surveys how word hyperbolic groups can harbor subgroups with exotic finiteness properties, including kernels of maps to $\mathbf{Z}$ that are of type $\mathcal{F}_n$ but not $\mathcal{F}_{n+1}$, and even fibers of type $\mathcal{F}$ that are not hyperbolic. It presents two parallel construction strategies: a complex-geometry route using holomorphic $1$-forms with isolated zeros and Albanese maps to produce the needed kernels, and a right-angled polytope route using cubulations and Bestvina–Brady Morse theory to obtain fibrations with exotic kernels. The complex-geometry method leverages forms with isolated zeros on Kähler manifolds, finite-to-one maps to tori, and Albanese immersions to realize the kernels; the polytope method builds explicit hyperbolic manifolds from a right-angled polytope, fibers them over the circle, and then caps boundaries to preserve CAT$(-1)$ structure while producing non-hyperbolic fiber groups. Together, these constructions answer longstanding questions about subgroups of hyperbolic groups and demonstrate the abundance of exotic finiteness phenomena in this setting.

Abstract

Gromov Hyperbolic groups have remarkable finiteness properties;for example those that are torsion-free are fundamental groups of finitecomplexes whose universal cover iscontractible (property~$F$). In this talk we will show thattheir subgroups can have exotic finiteness properties:there are hyperbolic groups containing finitely generated subgroups withintermediate finiteness properties (Llosa Isenrich and Py); there are hyperbolic groups containing finitely generated subgroups having theproperty~$F$ but which are not themselves hyperbolic (Italiano, Martelli, and Migliorini). This answersold questions about hyperbolic groups and their subgroups. The two mentioned results come from constructions of fibrations,the first in complex geometry and the second in hyperbolic geometry.We will describe the main points of these constructions.