Finite Stature in Artin groups

TL;DR

The paper develops and applies a finite-stature framework for graphs of groups to triangle Artin groups that split as graphs of free groups. By establishing that many such Artin groups admit monochrome-cycle preserving splittings, it proves finite stature with respect to the natural vertex-group subgroups, which then yields separability of finitely generated vertex-group subgroups and residual finiteness for a broad family of triangle Artin groups. The approach combines Stallings core theory, fiber-product intersections, and a color-encoded graph framework to control stabilizers along Bass–Serre trees, handling numerous parity and size cases of the labels M,N,P. This significantly extends known residually finite Artin groups and provides a robust method for proving residual finiteness in nonhyperbolic, non-virtually-special settings.

Abstract

We give criteria for a graph of groups to have finite stature with respect to its collection of vertex groups, in the sense of Huang-Wise. We apply it to the triangle Artin groups that were previously shown to split as a graph of groups. This allows us to deduce residual finiteness, and expands the list of Artin groups known to be residually finite.

Figures (15)

• Figure 1: Every length $6$ path in the Bass-Serre tree of $A*_CB$ where $[B:C] = 2$ is conjugate to the pictured path. The labels are the stabilizers. We note that two consecutive edges meeting at a $B$-vertex have the same stabilizers. See Lemma \ref{['lem:stabilizers as intersections']}. Algebraically, this also follows from the fact that $C^b = C$, since $[B:C] = 2$.
• Figure 2: The map $\phi:X_C\xrightarrow{\sigma}\overline{X}_C\xrightarrow{\iota} X_A$ when (1) none, (2) one, (3) two or (4) all of $M, N, P$ are even, respectively. Specifically, $M=2m$ or $2m+1$, $N=2n$ or $2n+1$, and $P=2p$ or $2p+1$. We use the convention where the edge labelled by a number $k$ is a concatenation of $k$ edges of the given color. The thickened edges in $X_{C}$ are the ones that get collapsed to a vertex in $\overline X_C$
• Figure 5: $(M,N,P) = (2m+1,4,4)$. The graph on the left is the fiber product $\overline Y_2 = \overline X_C\otimes_{X_A}\overline X_C$. The graph on the right is $\sigma\beta(Y_2)\otimes_{X_A} \overline Y_2$.
• Figure 6: $(M,N,P) = (2m+1,4,4)$. The vertical arrows are respectively: $\overline Y_2 \to \overline X_C$, $Y_2\to X_C$, $\beta(Y_2)\to X_C$, and $\sigma\cdot \beta(Y_2) \to \overline X_C$.
• Figure 7: $(M,N,P) = (2m+1, 2n+1, 2p+1)$. The graph on the left is the fiber product $\overline Y_2 = \overline X_C\otimes_{X_A}\overline X_C$. The graph on the right is $\sigma\beta(Y_2)\otimes_{X_A} \overline Y_2$.

Theorems & Definitions (71)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3: Stallings83
• Lemma 2.4: Stallings83
• Lemma 2.5: GMRS98
• proof
• Definition 3.1: HuangWiseStature19