Square-section braid groups and Higman-Neumann-Neumann extensions

TL;DR

The paper investigates the fundamental groups of discrete configuration spaces of hard squares, focusing on $B_{pq-2}(p\times q)$ and its topology. Using discrete configuration-space models and Farley–Sabalka gradient fields, it derives explicit simple commutator-relator presentations, with Betti-number parameters $\beta_1=(p-1)(q-1)+1$ and $\beta_2=\frac{(p^2+1)(q^2+1)-pq(2p+2q+3)+7(p+q-1)}{2}$, and proves torsion-free homology with $\operatorname{hdim}$ equal to $1$ for $p=q=3$ and $2$ otherwise. It then identifies $B_{2p-2}(p×2)$ as a RAAG and $B_{3p-2}(p×3)$ as an HNN extension of a RAAG, providing a Salvetti-complex viewpoint and a mapping-torus interpretation. The results reveal a cohesive picture linking RAAGs, HNN extensions, and discrete graph braid structures in the study of configuration spaces of hard squares, with implications for understanding the homotopy types of near-planar configuration spaces and their algebraic presentations.

Abstract

For positive integers $n$, $p$ and $q$ with $pq-n>0$, let $UC(n,p\times q)$ denote the configuration space of $n$ unlabelled hard unit squares in the rectangle $[0,p]\times[0,q]$, and let $B_n(p\times q)$ denote the corresponding fundamental group. It is known that, as $pq-n$ becomes large, $UC(n,p\times q)$ starts capturing homotopical properties of the classical configuration space of $n$ unlabelled pairwise-distinct points in the plane. At the start of this approximation process, $UC(pq-1,p\times q)$ is homotopy equivalent to a wedge of $(p-1)(q-1)$ circles, while the only other general families of spaces $UC(n,p\times q)$ known to be aspherical are $UC(n,p\times2)$ for $p\geq n$, and $UC(pq-2,p\times q)$. The fundamental groups of the former family are known to be responsible for the ``right-angled'' relations in Artin's classical braid groups. We prove that the fundamental groups of the latter family have a minimal presentation all whose relators are commutators. In particular, after explaining how $B_{2p-2}(p\times2)$ arises as the right-angled Artin group (RAAG) associated to a certain meta-edge, we show that $B_{3p-2}(p\times3)$ is a Higman-Neumann-Neumann extension of the RAAG associated to the corresponding meta-square. We provide a geometric interpretation of the latter fact in terms of Salvetti complexes.

Figures (5)

• Figure 1: The bipartite graph $E_k$. Edge thickness suggests "$v$-families" of edges. Since $u_i$ and $v_i$ are vertices of degree $i$, these families are dually reorganized by turning $E_k$ upside down.
• Figure 2: $B_{16}(6\times3)$ is the HNN extension of the indicated "square-type" RAAG with respect to the isomorphism $\varphi_6\colon F(x_1,x'_1)\to F(y_1,y'_1)$ given by $\varphi_6(x_1)=y_1$ and $\varphi_6(x'_1)=y'_1$. So, the relations $v_6x_1v_6^{-1}=y_1$ and $v_6x'_1v_6^{-1}=y'_1$ hold in $B_{16}(6\times3)$. Note that the subgroup generated by $x_1$ and $x'_1$ (by $y_1$ and $y'_1$, respectively) is free, as the subgraph generated by these two vertices has no edges.
• Figure 3: The first subgroup under conjugation is generated by $x_1$, $x_2$, $x'_1$ and $x'_2$, while the second one is generated by $y'_1$, $y'_2$, $y_1$ and $y_2$. They are again free due to the lack of edges between these vertices. The isomorphism $\varphi_7\colon F(x_1,x_2,x'_1,x'_2)\to F(y_1,y_2,y'_1,y'_2)$ is now given by $\varphi_7(x_i)=y_{3-i}$ and $\varphi_7(x'_i)=y'_{3-i}$, for $i=1,2$, so the stable generator $v_7$ is related to the rest of the generators via the relations $v_7x_iv_7^{-1}=y_{3-i}$ and $v_7x'_iv_7^{-1}=y'_{3-i}$$(1\leq i\leq2$).
• Figure 4: The domain of the isomorphism $\varphi_8$ defining the HNN extension is generated by $x_1,x_2,x_3,x'_1,x'_2,x'_3$, while the codomain is generated by the corresponding "$y$" elements. The isomorphism is determined by $\varphi_8(x_i)=y_{4-i}$ and $\varphi_8(x'_i)=y'_{4-i}$, for $1\leq i\leq3$, which correspond to relations $v_8x_iv_8^{-1}=y_{4-i}$ and $v_8x'_iv_8^{-1}=y'_{4-i}$ ($1\leq i\leq3$) in $B_{22}(8\times3)$. These relations together with the stable letter $v_8$ must then be added to the RAAG presentation of the indicated graph to yield a presentation for $B_{22}(8\times3)$. The two "unexpected" RAAG-type commutation relations $x_1x'_1=x'_1x_1$ and $y_1y'_1=y'_1y_1$ correspond to the two thick edges. The former one, for instance, is forced from the fact that $v_8x_1v_8^{-1}=y_3$ commutes with $v_8x'_1v_8^{-1}=y'_3$. In particular, neither the domain nor the domain of $\varphi_8$ are free; they are instead RAAGs associated to the induced subgraphs.
• Figure 5: The graph $\Gamma_{6,4}$. Fill in the 15 loops with solid squares to get the complex $\operatorname{Puz}_{6,4}$.

Theorems & Definitions (27)

• Remark 1.1
• Definition 1.2: Section 2 of MR4096335
• Proposition 1.3: AGK, MR4640135 and MR4298668
• Definition 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Definition 2.1
• Theorem 2.2
• Corollary 2.3