Conjugacy invariants and rigidity in Garside groups: a uniformity phenomenon

TL;DR

This work investigates when the sliding circuits sizes $|SC(x^n)|$ of rigid elements in Garside groups become periodic as $n$ grows, showing that boundedness implies periodicity and that the period is uniformly bounded by group-dependent quantities. The authors develop a robust framework linking rigid elements, conjugacy graphs, and power maps, proving key lattice-type results: rigid powers propagate in controlled ways and primitive rigid powers are finitely many. They establish the main conjecture for circular Garside groups and for the dual 4-strand braid group, providing explicit bounds: in circular groups, the period is 1 with a sharp bound $rac{m}{m\wedge \ell}$ on rigid powers; in $\mathcal{B}_4^*$ the period divides $\mathrm{lcm}(1,\dots,27)$. These results illuminate a uniformity phenomenon across a fixed Garside structure and connect to broader geometric and group-theoretic contexts. The work combines deep algebraic structure with extensive computational evidence to advance the conjugacy problem and its invariants in Garside theory.

Abstract

Consider an element~$x$ of a Garside group which is rigid in the sense of Garside-theory. Let $SC(x)$ be the set of rigid conjugates of~$x$ -- this is a well-known characteristic subset of the conjugacy class of~$x$. We present computational evidence that the sequence $( |SC(x^n)| )_{n\in\mathbb N}$ is not only bounded, but in fact periodic, and that there is a bound on the length of the period which depends only on the underlying group and its Garside structure. We prove this result in the special case of the circular Garside groups, including the 2-generator Artin groups with their classical and dual structures (where we prove that the sequence is always constant), and in the case of the dual $4$-strand braid group.

Figures (10)

• Figure 1: For $x=21|12|2132$, the figure shows on the left the conjugacy graph of $x$ (only one vertex), and on the right the conjugacy graph of $x^2$.
• Figure 2: The conjugation of $x^2$ by $1$ for $x=21|12|2132$, seen in the Cayley graph. This diagram can be periodically extended into a bi-infinite strip. This type of diagram is called a domino diagram.
• Figure 3: A domino diagram showing the calculation of the normal form of $c^{-1}x\,c$ if $\inf(x)=0$. It is a non-obvious fact that the word $y_1\,y_2\,y_3$ obtained by this calculation is in normal form. Also, under the hypothesis that both $x$ and $c^{-1}x\, c$ are rigid, it is true but non-obvious that $d_0=d_3$ and $c_0=c_3=c$, meaning that $y_1\, y_2\, y_3=c^{-1} x\, c$, as desired.
• Figure 4: A domino diagram showing the calculation of the normal form of $c^{-1}x\,c$, in the case $\inf(x)\neq 0$. Again, the word $\Delta^k y_1\,y_2\,y_3$ obtained by this calculation is in normal form. Also, under the hypothesis that both $x$ and $c^{-1}x\, c$ are rigid, $d_0=d_3$ and $c_0=c_3=c$, meaning that $\Delta^k y_1\, y_2\, y_3=c^{-1} x\, c$.
• Figure 5: The conjugacy graph of $x^{12}$. However, for the sake of clarity, not all black and gray arrows are shown but only the minimal ones, i.e. only those arrows that cannot be obtained as the composition of two or more arrows of the same color.

Theorems & Definitions (52)

• Conjecture 1.1
• Remark 1.3
• Example 1.5
• Conjecture 1.7
• Proposition 1.9
• Theorem 1.10
• Theorem 1.11
• Remark 1.12
• Remark 1.13
• Definition 2.1