[Paper Review] Irreducible lattices fibring over the circle

Sam Hughes|arXiv (Cornell University)|Jan 17, 2022

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper establishes the existence of irreducible uniform lattices in products of $<\mathrm{CAT}(0)\u003e$ spaces that virtually algebraically fiber over the circle, using a construction based on Bestvina–Brady groups and RAAGs. It proves that for lattices in $\mathrm{Isom}(\mathbb{E}^n) \times \mathrm{Aut}(\mathcal{T})$, virtual algebraic fibration implies reducibility, but constructs explicit irreducible examples with kernels of finite type via BNSR invariants.

ABSTRACT

We investigate the Bieri--Neumann--Strebel--Renz (BNSR) invariants of irreducible uniform lattices. In the case of a direct product of a tree and a Euclidean space we show that vanishing of the BNSR invariants for all finite-index subgroups of a given uniform lattice is equivalent to irreducibility. On the other hand we construct irreducible uniform lattices which admit maps to the integers whose kernels' finiteness properties are determined by the finiteness properties of certain Bestvina--Brady groups.

Motivation & Objective

• To determine whether virtual algebraic fibration of irreducible lattices in products of CAT(0) spaces implies reducibility.
• To construct explicit examples of irreducible lattices that virtually fiber over the circle with kernels of finite type.
• To extend the theory of BNSR invariants to lattices in products of isometry groups of minimal unbounded CAT(0) spaces.
• To clarify the relationship between the finiteness properties of kernels of characters and the structure of the underlying lattice.

Proposed method

• Constructing a lattice $\Gamma_L$ as a quotient of a right-angled Artin group (RAAG) $A_L$ acting on a CAT(0) cube complex $\widetilde{X}_L$.
• Defining a character $\psi \in H^1(\Gamma_L; \mathbb{R})$ via a height function on $\widetilde{X}_L$ induced by a character $\phi \in H^1(A_L; \mathbb{R})$.
• Using Brown’s criterion to verify that $\ker(\psi)$ is of type $\mathsf{F}_\infty$ (and $\mathsf{FP}_\infty$) when $\ker(\phi)$ is of type $\mathsf{F}_\infty$.
• Applying known results on BNSR invariants of RAAGs to determine when $\ker(\phi)$ is of finite type, particularly using the link condition on the flag complex $L^*$.
• Proving that $\Gamma_L$ is irreducible and virtually torsion-free, ensuring the existence of a finite-index torsion-free subgroup with integral character.
• Establishing that $\ker(\psi)$ is of type $\mathsf{F}$ when restricted to a finite-index torsion-free subgroup, thus yielding a virtual fibration.

Experimental results

Research questions

• RQ1 Does virtual algebraic fibration of an irreducible uniform lattice in $\mathrm{Isom}(\mathbb{E}^n) \times \mathrm{Aut}(\mathcal{T})$ imply reducibility?
• RQ2 Can irreducible lattices in products of CAT(0) spaces admit characters with kernels of finite type?
• RQ3 What is the role of BNSR invariants in determining the finiteness properties of kernels of characters on such lattices?
• RQ4 How do the finiteness properties of kernels of characters on RAAGs transfer to associated lattices in product groups?
• RQ5 Is there a construction of an irreducible lattice that virtually fibers over the circle with a finitely presented kernel?

Key findings

• The paper constructs an explicit example of an irreducible uniform lattice in $\mathrm{PSL}_2(\mathbb{R}) \times \mathrm{PSL}_2(\mathbb{Q}_2)$ that virtually algebraically fibers over the circle.
• For lattices in $\mathrm{Isom}(\mathbb{E}^n) \times \mathrm{Aut}(\mathcal{T})$, virtual algebraic fibration implies reducibility, establishing a converse to the general fibration question.
• The kernel of the character $\psi$ on the constructed lattice $\Gamma_L$ is of type $\mathsf{F}_\infty$, and after passing to a finite-index torsion-free subgroup, the kernel is of type $\mathsf{F}$.
• The finiteness properties of $\ker(\psi)$ are determined by the BNSR invariants of the associated RAAG $A_L$, via the equivalence in Theorem 5.5.
• The construction relies on a flag complex $L$ of girth at least 4 with contractible links and non-trivial structure, ensuring $A_L$ is irreducible and $\ker(\phi)$ is of type $\mathsf{F}$.
• The existence of such a lattice confirms that irreducible lattices can fiber over the circle, answering the main question affirmatively.