[Paper Review] All finitely presented groups are QSF

Valentin Poénaru|arXiv (Cornell University)|Sep 25, 2014

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper proves that all finitely presented groups are quasi-geometrically simply connected (QSF), establishing a fundamental property in geometric group theory. Using a 3-dimensional singular handlebody construction and a high-dimensional thickening via the $S_u$ functor, the authors show that the universal cover of a certain manifold model for any finitely presented group admits a geometrically simply connected structure, thereby confirming the QSF property for all such groups.

ABSTRACT

This is the third and last of three papers containing the complete proof that all finitely presented groups are QSF.

Motivation & Objective

• To establish that every finitely presented group satisfies the QSF (quasi-simplex-filling) property, a central conjecture in geometric group theory.
• To construct a universal cover model for any finitely presented group that is geometrically simply connected (GSC), implying QSF.
• To resolve the local finiteness obstruction in high-dimensional cell-complexes by introducing compensating 2-handles to restore finiteness while preserving group action and topology.
• To demonstrate that the $S_u\widetilde{M}(\Gamma)$ complex, built from a 3D singular presentation, is GSC, which implies the QSF property for $\Gamma$.
• To show that the QSF property is equivalent to the existence of a smooth closed manifold with fundamental group $\Gamma$ whose universal cover is GSC.

Proposed method

• Construct a 3-dimensional singular handlebody $M(\Gamma)$ with $\pi_1(M(\Gamma)) = \Gamma$, using a 3D presentation to avoid infinite accumulations in the universal cover.
• Apply the $S_u$ functor to thicken the universal cover $\widetilde{M}(\Gamma)$ into a high-dimensional cell-complex $S_u\widetilde{M}(\Gamma)$ of dimension $N+4$, ensuring $\Gamma$-action and GSC structure.
• Introduce a system of compensating 2-handles in the $N+4$-dimensional complex to restore local finiteness, resulting in a singular cell-complex rather than a smooth manifold.
• Use a $\Gamma$-equivariant process involving proper Whitehead dilatation and infinite handle additions to produce a co-compact subcomplex $\Theta^3(\text{co-compact}) \subset \Theta^3(fX^2)$.
• Replace immersive maps with simplicial maps via cone constructions on components $C_i$, allowing non-immersive but simply-connected source complexes to maintain control over fundamental group.
• Iteratively disentangle the complex $K$ from preimages of singular sets $\pi^{-1}A_i$ and $\pi^{-1}B_j$ via perturbations and cone-based splittings, ultimately achieving a clean embedding with $K \cap \Theta_0^3 = S_0 \cap k$.

Experimental results

Research questions

• RQ1 Does every finitely presented group $\Gamma$ satisfy the QSF property?
• RQ2 Can a geometrically simply connected (GSC) universal cover be constructed for any finitely presented group?
• RQ3 Is the $S_u\widetilde{M}(\Gamma)$ complex, built from a 3D singular presentation, geometrically simply connected?
• RQ4 Can local finiteness be restored in high-dimensional thickenings of universal covers without losing the GSC property?
• RQ5 Is the QSF property equivalent to the existence of a smooth closed manifold $M$ with $\pi_1(M) = \Gamma$ and GSC universal cover?

Key findings

• All finitely presented groups $\Gamma$ satisfy the QSF property, confirming a long-standing conjecture in geometric group theory.
• The $S_u\widetilde{M}(\Gamma)$ complex is geometrically simply connected (GSC), as established in Theorem B of the preceding paper [39], which is central to this proof.
• The construction of $S_u\widetilde{M}(\Gamma)$ as a thickening of a 4D complex $\Theta^4(\Theta^3(fX^2), \mathcal{R})$ with compensating 2-handles ensures local finiteness while preserving the GSC structure.
• The $\Gamma$-action on $\Theta^3(fX^2)$ is not cocompact, but a $\Gamma$-invariant co-compact subcomplex $\Theta^3(\text{co-compact})$ exists via proper dilatation and handle addition.
• The final complex $K$ is disentangled from all singular sets $\pi^{-1}A_i$ and $\pi^{-1}B_j$, achieving $K \cap \Theta_0^3 = S_0 \cap k$, which satisfies the required diagram (4.2).
• The QSF property is equivalent to the existence of a smooth closed manifold $M$ with $\pi_1(M) = \Gamma$ and GSC universal cover, as shown via results of Funar and Otera.