Colored Stallings graphs and Counterexamples to Stallings equalizer conjecture

Jialin Lei; Teng Zhang

paper

Colored Stallings graphs and Counterexamples to Stallings equalizer conjecture

arXiv:2604.24502

Abstract

The famous Stallings equalizer conjecture has remained open for more than 40 years, which states that, for any free group $F_n$ of rank $n\ge 2$, any free group $F$, and any two monomorphisms $g, h : F_{n} \to F,$ the equalizer $\Eq(g,h)=\{w\in F_n\mid g(w)=h(w)\}$ satisfies $\rk \Eq(g,h)\le n.$ The only known case is $n = 2$ , due to A. D. Logan in 2022. By introducing the notion of colored Stallings graphs, we show that for every integer $n\ge 2$ there exist monomorphisms $g, h : F_{n} ⟶ F_{2}$ such that$\rk\Eq(g,h)\ge 2n-2.$ This disproves Stallings equalizer conjecture for $n \geq 3$ .

v2, 15 pages, this is the submitted version; v1, 10 pages, all comments are welcome!