Groups with exotic finiteness properties from complex Morse theory
Claudio Llosa Isenrich, Pierre Py
TL;DR
The paper develops a systematic framework, grounded in complex Morse theory and BNSR-invariants, for producing groups with exotic finiteness properties (types $\mathscr{F}_{n}$ vs ${\rm FP}_n(\mathbb{Q})$) from complex-analytic geometry. By leveraging maps to complex tori, Stover–Toledo ball quotients, fibre-product constructions, and iterated Kodaira fibrations, it constructs both Kähler and non-Kähler groups whose kernels in surjections onto $\mathbb{Z}$ or $\mathbb{Z}^2$ have refined finiteness profiles (e.g., $\mathscr{F}_{n-1}$ but not ${\rm FP}_n(\mathbb{Q})$). The work also produces nonnormal subgroups via fibre products and provides new proofs of finiteness phenomena (e.g., Kochloukova–Vidussi) in the realm of complex geometry, illustrating the power of complex Morse methods beyond classical examples. Overall, the results significantly extend the supply of groups with exotic finiteness properties arising from geometric constructions, with implications for hyperbolic, Kähler, and non-Kähler contexts, and raise questions about higher-rank abelian quotients and more general fibre-product setups.
Abstract
Recent constructions have shown that interesting behaviours can be observed in the finiteness properties of Kähler groups and their subgroups. In this work, we push this further and exhibit, for each integer $k$, new hyperbolic groups admiting surjective homomorphisms to $\mathbb{Z}$ and to $\mathbb{Z}^{2}$, whose kernel is of type $\mathscr{F}_{k}$ but not of type $\mathscr{F}_{k+1}$. By a fibre product construction, we also find examples of nonnormal subgroups of Kähler groups with exotic finiteness properties.