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18.89 (2014)

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Consider the set of balanced presentations $\langle x_1, \dots, x_n \mid r_1, \dots, r_n \rangle$ with fixed $n$ generators $x_1, \dots, x_n$. By definition, the group $\text{AC}_n$ of Andrews–Curtis moves on this set of balanced presentations is generated by the Nielsen transformations together with conjugations of relators. Is $\text{AC}_n$ finitely presented?

Contributor: J. Swan, A. Lisitsa

Progress

*The group $\text{AC}_2$ is not finitely presented (S. Krstić, J. McCool, J. London Math. Soc. (2), 56, no. 2 (1997), 264–274; also V. A. Roman’kov, Preprint, 2023, https://arxiv.org/pdf/2305.11838.pdf), while the groups $\text{AC}_n$, $n \geqslant 3$, are finitely presented, which follows from (G. Kiralis, S. Krstić, J. McCool, Proc. London Math. Soc. (3), 73, no. 3 (1996), 481–720) (M. Ershov, Letter of 16 October 2025, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2025/11/kourovka1889.pdf).

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