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Perfect circle-valued Morse functions on hyperbolic 6-manifolds

Giovanni Italiano, Matteo Migliorini

TL;DR

The authors address the obstruction to fibrations in even dimensions by constructing a cusped hyperbolic $M^6$ that carries a perfect circle-valued Morse function $f:M^6\to S^1$, with all critical points of index $3$. The approach blends the generalized Bestvina–Brady framework on barycentric subdivisions of a cubulation built from the polytope $P^6$ (via good/bad faces and inherited state) with an explicit cusp compactification and smoothing, yielding a topological model where $f$ is perfect. A key outcome is a finitely presented subgroup of a hyperbolic group that is of type $\mathcal{F}_2$ but not $\mathcal{F}_3$, illustrating nontrivial higher finiteness properties in this setting. The work advances higher-dimensional fibrations in hyperbolic geometry, clarifies the role of combinatorial cubulations in geometric group theory, and opens pathways to analogous results in dimensions $7$ and $8$ and beyond.

Abstract

We build the first example of a hyperbolic 6-manifold that admits a perfect circle-valued Morse function, which can be considered as the analogue of a fibration over the circle for manifolds with non-vanishing Euler characteristic. As a consequence, we obtain a new example of a subgroup of a hyperbolic group which is of type $\mathcal{F}_2$ but not $\mathcal{F}_3$.

Perfect circle-valued Morse functions on hyperbolic 6-manifolds

TL;DR

The authors address the obstruction to fibrations in even dimensions by constructing a cusped hyperbolic that carries a perfect circle-valued Morse function , with all critical points of index . The approach blends the generalized Bestvina–Brady framework on barycentric subdivisions of a cubulation built from the polytope (via good/bad faces and inherited state) with an explicit cusp compactification and smoothing, yielding a topological model where is perfect. A key outcome is a finitely presented subgroup of a hyperbolic group that is of type but not , illustrating nontrivial higher finiteness properties in this setting. The work advances higher-dimensional fibrations in hyperbolic geometry, clarifies the role of combinatorial cubulations in geometric group theory, and opens pathways to analogous results in dimensions and and beyond.

Abstract

We build the first example of a hyperbolic 6-manifold that admits a perfect circle-valued Morse function, which can be considered as the analogue of a fibration over the circle for manifolds with non-vanishing Euler characteristic. As a consequence, we obtain a new example of a subgroup of a hyperbolic group which is of type but not .

Paper Structure

This paper contains 19 sections, 34 theorems, 8 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bestvina-Brady Morse theory
  4. Jankiewicz, Norin, and Wise's algorithm
  5. Bad faces and state inheritance
  6. The barycentric subdivision
  7. Ascending and descending face links
  8. The coface link
  9. The polytope $P^6$
  10. Potyagailo-Vinberg polytopes
  11. The polytope $P^5$
  12. The polytope $P^6$
  13. The set of moves
  14. States
  15. Computing the states
  16. ...and 4 more sections

Key Result

Theorem 1

There exists a hyperbolic $6$-manifold $M^6$ that admits a perfect circle-valued Morse function.

Figures (11)

  • Figure 1: The two possible orientations on a square producing a $1$-cocycle, up to rotation.
  • Figure 2: On the left, a square $Q$ with its edges oriented; the number on the vertices are the values of the lift of $f^{(1)}$ to $\mathbb R$. On the right, the values of the lift $\tilde{f} \colon \mathop{\mathrm{sd}}\nolimits Q \to \mathbb R$.
  • Figure 3: On the left, a monochromatic square; on the right, a monochromatic cube. The numbers on the vertices denote the values of $\tilde{f}$ on the vertices.
  • Figure 4: On the left, the product of a bad square with an interval. On the right, the ascending (blue) and descending (red) face links. The descending link is a join of the face links of a bad square and of an interval, while the ascending link only collapses on a join (drawn with a thick blue line).
  • Figure 5: In this picture, we represented a cube $Q'$ of the cubulation, which is a cofacet of another cube $Q$ (the bottom edge). The blue vertex is the barycentre of $P_{v}$, and the blue edge is dual to a facet $F$ of $P_{v}$ with status $I$. When computing the coface link at the red vertex, which is the barycentre of a face $P'$ of $P_{v}$, we need to know the orientation of the red edge, corresponding to the status of the facet $F'=F\cap P'$ of $P'$. In the picture on the left, the square is coherent, so the orientation of the red edge coincides with the orientation of the blue one: the status of $F'$ coincides with the status of $F$. Vice versa, on the right, we have a bad square: in this case, the red edge is always oriented outwards, so the status of $F'$ is always $O$, independently of the status of $F$.
  • ...and 6 more figures

Theorems & Definitions (95)

  • Theorem 1
  • Corollary 2
  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: BestvinaBradyMorseTheory
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 85 more