[Paper Review] Hyperbolic groups with almost finitely presented subgroups

Robert Kropholler, Federico Vigolo|arXiv (Cornell University)|Sep 27, 2018

Geometric and Algebraic Topology17 references4 citations

TL;DR

This paper constructs the first known examples of hyperbolic groups containing subgroups that are of type $FP_2$ but not finitely presented, using a novel construction based on CAT(0) cube complexes and group presentations with infinite relators. The key contribution is a hyperbolic group $G$ with a homomorphism $\phi: G \to \mathbb{Z}$ such that $\ker(\phi)$ is of type $FP_2$ yet not finitely presented, demonstrating that hyperbolicity does not imply finite presentability for $FP_2$ subgroups beyond cohomological dimension 2.

ABSTRACT

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups which are finitely presented but not of type $FP_3$.

Motivation & Objective

• To construct subgroups of hyperbolic groups with strong homological finiteness properties but without finite presentability.
• To challenge the conjecture that $FP_2$ subgroups of hyperbolic groups must be finitely presented.
• To explore the limits of embedding $FP_2$ groups into hyperbolic groups, particularly in cohomological dimension 3.
• To generate uncountably many $FP_2$ groups without $BS(m,n)$ subgroups or infinite torsion, avoiding known obstructions to hyperbolic embedding.

Proposed method

• Using a specific finitely generated, infinitely presented group $H$ with presentation $\langle S \mid U \rangle$, where $S$ is finite and $U$ is infinite.
• For each subset $Z \subseteq U$, defining a quotient group $H(Z) = \langle S \mid Z \rangle$ to generate a family of groups.
• Proving that only countably many $H(Z)$ are isomorphic to each other, ensuring uncountably many distinct isomorphism classes.
• Showing that uncountably many of these $H(Z)$ are of type $FP_2$ via homological finiteness criteria.
• Constructing a hyperbolic group $G$ via CAT(0) cube complexes from tripartite flag complexes $\Gamma_A$ and $\Gamma_B$, using the product structure $\prod_{i=1}^3 A_i * B_i$.
• Defining a maximal subcomplex $\mathbf{X}_{\Gamma_A,\Gamma_B}$ whose fundamental group yields the desired hyperbolic group $G$, and constructing a homomorphism $\phi: G \to \mathbb{Z}$ with $\ker(\phi)$ of type $FP_2$ but not finitely presented.

Experimental results

Research questions

• RQ1 Can there exist a subgroup of a hyperbolic group that is of type $FP_2$ but not finitely presented?
• RQ2 Is the property of being $FP_2$ equivalent to finite presentability for subgroups of hyperbolic groups?
• RQ3 Are there uncountably many $FP_2$ groups without $BS(m,n)$ subgroups or infinite torsion, which could potentially embed into hyperbolic groups?
• RQ4 What are the obstructions to embedding $FP_2$ groups into hyperbolic groups, and can these be overcome?

Key findings

• The paper constructs a hyperbolic group $G$ and a homomorphism $\phi: G \to \mathbb{Z}$ such that $\ker(\phi)$ is of type $FP_2$ but not finitely presented, providing the first such example.
• The kernel $\ker(\phi)$ is not of type $FP_3$, showing that the phenomenon is specific to $FP_2$ and not extendable to higher finiteness properties.
• Among the groups $H(Z) = \langle S \mid Z \rangle$, uncountably many are of type $FP_2$, despite only countably many being isomorphic to each other.
• The constructed $FP_2$ groups avoid all known obstructions to embedding into hyperbolic groups, such as containing $BS(m,n)$ subgroups or infinite torsion subgroups.
• The construction relies on CAT(0) cube complexes with tripartite flag complexes, where the link structure ensures non-positive curvature and hyperbolicity.
• The method demonstrates that the equivalence between $FP_2$ and finite presentability fails in hyperbolic groups of cohomological dimension 3, unlike in dimension 2.