Virtual Cactus Group is Virtually Special Compact

TL;DR

The paper proves that the virtual cactus group $vC_n$ is virtually special compact by modeling $PvC_n$ as the fundamental group of the NPC cube complex $\widehat{D}_n$ and constructing a finite cover $M_n$ of $\widehat{D}_n$ that is a compact special cube complex. The construction uses planar forests labeled by leaves and a controlled gluing along hyperplanes to produce a finite covering, with degree $2^{2^n-n-1}$, and identifies the kernel of a homomorphism $PvC_n\to(\mathbb{Z}/2\mathbb{Z})^L$ as $\pi_1(M_n)$. A detailed hyperplane analysis shows all pathologies (self-intersections, self-osculation, inter-osculation) disappear in the cover, establishing that $M_n$ is special. Consequently, $vC_n$ is virtually the fundamental group of a compact special cube complex, implying $\mathbb{Z}$-linearity for the virtual cactus group and providing a geometric route to linearity via specialness. The work leverages the combinatorics of planar forests and hyperplane types to control cube-complex embeddings and intersections.

Abstract

We prove that the virtual cactus group has a finite index subgroup that is the fundamental group of a compact special cube complex.

Figures (10)

• Figure 1: A partial sketch of the cube complex $D_3$ using subcubes.
• Figure 2: Reflections $Z_i \cdot \tau$ in $\widehat{D}_3$
• Figure 3: As a complex made of subcubes, a 1-cube is made from gluing two 1-subcubes at a point. As a complex made of cubes, a 1-cube is made from identifying two 1-cubes in reversed orientation.
• Figure 4: The two types of 2-subcubes in $\widehat{D}_n$
• Figure 5: An example of the argument for case 3 when $e_2$ lies above $e_1$. We choose not draw the trees in the forests with a single internal edge.

Theorems & Definitions (27)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Example 2.1
• Remark 2.2
• Example 2.3
• Remark 2.4
• Lemma 3.1
• proof
• Remark 3.2