Colored Stallings graphs and Counterexamples to Stallings equalizer conjecture
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909029315379200 |
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| author | Lei, Jialin Zhang, Teng |
| author_facet | Lei, Jialin Zhang, Teng |
| contents | 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 $g,h:F_n\longrightarrow F_2$ such that$\rk\Eq(g,h)\ge 2n-2.$ This disproves Stallings equalizer conjecture for $n\ge 3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_24502 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Colored Stallings graphs and Counterexamples to Stallings equalizer conjecture Lei, Jialin Zhang, Teng Group Theory 20E05, 20E07, 20F65 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 $g,h:F_n\longrightarrow F_2$ such that$\rk\Eq(g,h)\ge 2n-2.$ This disproves Stallings equalizer conjecture for $n\ge 3$. |
| title | Colored Stallings graphs and Counterexamples to Stallings equalizer conjecture |
| topic | Group Theory 20E05, 20E07, 20F65 |
| url | https://arxiv.org/abs/2604.24502 |