The Yang-Baxter Equation and Characteristic Finite Simple Quotients of the Free Group of Rank $2$

TL;DR

This work advances the understanding of how the free group $F_2$ admits characteristic finite quotients of both alternating and arbitrarily large Lie type, by bridging braid group actions, Yang–Baxter solutions, and quantum-group representations. The core strategy uses a novel equivariant quandle to induce a $B_4$-action on structured orbit sets, enabling primitive-action analysis that yields alternating quotients, and, separately, Zariski-dense quantum representations of $B_4$ to realize infinitely many large-rank quotients via strong approximation. The two main results are: (i) infinitely many characteristic quotients of $F_2$ are alternating groups $A_n$ for infinitely many $n$, and (ii) characteristic quotients of arbitrarily large Lie rank, specifically $PSL_{inom{n}{2}}(\mathbb{F}_q)$ for suitable $q$, arise from Zariski-dense representations; these extend prior Baby Wiegold-type observations. The paper also refines the residual-finiteness landscape by proving that $Aut(F_2)$ is fully residually finite almost-simple under the constructed quotients, and frames conditional statements about more general $Aut(F_n)$. Overall, the results provide new non-linear, characteristic finite quotients of free groups with potential implications for dynamics, 3-manifold covers, and the structure of automorphism groups of free groups.

Abstract

We show that infinitely many alternating groups arise as quotients of the free group of rank 2, with kernel a characteristic subgroup. We also show that such simple quotients exist of arbitrarily large Lie rank. This resolves two questions posed by arXiv:2308.14302

Theorems & Definitions (100)

• Conjecture 1.1: The "Baby Wiegold Conjecture"
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Corollary 1.5
• Definition 1.6
• Theorem 1.7
• Remark 1.8
• Definition 2.1
• Definition 2.2