[Paper Review] Geometric simple connectivity and finitely presented groups

Valentin Poénaru|arXiv (Cornell University)|Apr 16, 2014

Geometric and Algebraic Topology19 references6 citations

TL;DR

This paper establishes a 2-dimensional representation theorem for finitely presented groups, proving that every such group admits a geometrically simply connected (GSC) 2-complex with a $ ilde{M}( ho)$-equivariant, essentially surjective, nondegenerate simplicial map. The key contribution is the construction of a uniformly bounded zipping process, which implies the GSC theorem and completes the proof of the QSF property for all finitely presented groups via a geometric, group-theoretic approach using singular 3-manifolds and controlled topology at infinity.

ABSTRACT

This is the second of a Series of three papers, the first one published in Geom Dedicata 167 p. 91-121 (2013), proving that all finitely presented groups are QSF.

Motivation & Objective

• To establish a 2-dimensional representation theorem for any finitely presented group $\Gamma$, extending prior 3D results.
• To prove that the universal cover $\widetilde{M}(\Gamma)$ is geometrically simply connected (GSC), a key step toward proving the QSF property.
• To construct a $\Gamma$-equivariant, essentially surjective, nondegenerate simplicial map $f: X^2 \to \widetilde{M}(\Gamma)$ with discrete, mortal singularities.
• To ensure a uniformly bounded zipping length for the singular double point sets, enabling control at infinity.
• To complete the proof of the QSF conjecture for finitely presented groups using geometric and topological methods in singular 3-manifolds.

Proposed method

• Construct a 2-dimensional, locally finite, geometrically simply connected (GSC) simplicial complex $X^2$ as a skeleton of the 3D representation space from prior work.
• Define a nondegenerate simplicial map $f: X^2 \to \widetilde{M}(\Gamma)$ such that $f$ is $\Gamma$-equivariant and essentially surjective.
• Introduce the equivalence relations $\Psi(f) \subset \Phi(f)$ on $X^2$, where $\Psi(f)$ is the smallest relation making $X^2/\Psi(f) \to \widetilde{M}(\Gamma)$ immersive.
• Use zipping processes to eliminate mortal singularities, relying on controlled homotopies and singular disks in the boundary of a compactified manifold.
• Apply a log homotopy argument to avoid a compact set $K \subset \partial S'_u(M(\Gamma)-H)_{\rm II}$, ensuring the existence of embedded disks avoiding $K$.
• Leverage $\pi_1$-injectivity in a sub-lemma to construct singular disks in a cut locus, enabling the construction of uniformly bounded zipping paths.

Experimental results

Research questions

• RQ1 Can every finitely presented group $\Gamma$ be represented via a 2-dimensional, $\Gamma$-equivariant, essentially surjective simplicial map into $\widetilde{M}(\Gamma)$ with discrete singularities?
• RQ2 Does the existence of such a 2D representation imply that $\widetilde{M}(\Gamma)$ is geometrically simply connected (GSC)?
• RQ3 Can the zipping process for double points be uniformly bounded in length, independent of the group element?
• RQ4 Is the boundary of the universal cover at infinity controlled in such a way that null-homotopic curves can be filled without intersecting a compact obstruction?
• RQ5 Does the absence of $\pi_1^\infty$-obstructions in the compactified boundary imply the GSC property for $\widetilde{M}(\Gamma)$?

Key findings

• A 2-dimensional representation $f: X^2 \to \widetilde{M}(\Gamma)$ exists for every finitely presented group $\Gamma$, with $X^2$ locally finite and $\operatorname{Sing}(f)$ discrete.
• The map $f$ admits a free, $\Gamma$-equivariant action on $X^2$, ensuring the group acts geometrically on the source complex.
• There exists a uniform bound $M < \infty$ on the zipping length for all double point sets, ensuring controlled topology at infinity.
• The main lemma (6.5) establishes that a null-homotopic curve $\Lambda_{j_n}$ in $\partial S'_u(M(\Gamma)-H)_{\rm II}$ bounds a singular disk avoiding a compact set $K$, via a log homotopy and $\pi_1$-injectivity.
• The compactness lemma (4.7) is proven, implying that the GSC theorem (2.3) holds: $S_u\widetilde{M}(\Gamma)_{\rm II} \in \text{GSC}$.
• The proof confirms that the GSC property of $\widetilde{M}(\Gamma)$ implies the QSF property for $\Gamma$, completing the series of papers on the QSF conjecture.