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Cleanliness versus Specialness

Kasia Jankiewicz

TL;DR

The paper addresses whether every geometrically clean graph of finite rank free groups has a virtually compact special fundamental group. It answers in the negative by constructing a geometrically clean graph of graphs $X(\Theta)$ whose fundamental group $G$ embeds as an index-$6$ subgroup of the Artin group $A_{2,3,\infty}$, and hence is not virtually cocompact cubulated or virtually compact special. It then generalizes the construction to the odd-$n$ family $A_{2,n,\infty}$, producing an infinite collection of geometrically clean graphs of free groups that likewise fail to be virtually compact special. Consequently, the class of groups that virtually split as $\mathcal{VH}$-clean graphs of finite rank free groups is strictly contained in the class of geometrically clean graphs of finite rank free groups, clarifying the landscape of cleanliness notions and their link to virtual cubulation.

Abstract

We show that the fundamental group of a geometrically clean graph of finite rank free groups does not need to be virtually compact special, answering a question of Wise. This implies that the class of the virtually VH-clean graphs of finite rank free groups is a proper subclass of the class of virtually geometrically clean graphs of finite rank free groups.

Cleanliness versus Specialness

TL;DR

The paper addresses whether every geometrically clean graph of finite rank free groups has a virtually compact special fundamental group. It answers in the negative by constructing a geometrically clean graph of graphs whose fundamental group embeds as an index- subgroup of the Artin group , and hence is not virtually cocompact cubulated or virtually compact special. It then generalizes the construction to the odd- family , producing an infinite collection of geometrically clean graphs of free groups that likewise fail to be virtually compact special. Consequently, the class of groups that virtually split as -clean graphs of finite rank free groups is strictly contained in the class of geometrically clean graphs of finite rank free groups, clarifying the landscape of cleanliness notions and their link to virtual cubulation.

Abstract

We show that the fundamental group of a geometrically clean graph of finite rank free groups does not need to be virtually compact special, answering a question of Wise. This implies that the class of the virtually VH-clean graphs of finite rank free groups is a proper subclass of the class of virtually geometrically clean graphs of finite rank free groups.

Paper Structure

This paper contains 3 sections, 6 theorems, 2 equations, 4 figures.

Table of Contents

  1. Introduction and background
  2. The construction of the graph of groups $G(\Theta)$
  3. An infinite family of examples

Key Result

Theorem 3

Every group which splits as a $\mathcal{VH}$-clean graph of finite rank free groups is virtually compact special.

Figures (4)

  • Figure 1: A geometrically clean graph of graphs $X(\Theta)$, whose fundamental group $G(\Theta)$ is not virtually compact special.
  • Figure 2: On the left: a presentation complex of $A_{2,3,\infty}$. On the right: its double cover whose fundamental group is the kernel of a homomorphism to $\mathbb Z/2$.
  • Figure 3: The group $\pi_1 \hat{Y}$ splits as an amalgamated product $F_2*_{F_3}F_2$ with the two maps $F_3\to F_2$ induced by the graph maps above.
  • Figure 4: Geometrically clean graph of free groups, whose vertex group are copies of $F_8$, and edge groups are copies of $F_3$, whose fundamental group embeds as an index $14$ subgroup of the Artin group $A_{2,7,\infty}$.

Theorems & Definitions (8)

  • Definition 1
  • Theorem 3: HaglundWiseSpecial
  • Theorem 4
  • Corollary 5
  • Proposition 7
  • Proposition 8
  • proof
  • Corollary 9