Robert H. Gilman ; Alexei G. Myasnikov - Andrews-Curtis groups

gcc:15972 - journal of Groups, complexity, cryptology, July 4, 2025, Volume 16, Issue 1, Special issue in memory of Ben Fine - https://doi.org/10.46298/jgcc.2025..15972
Andrews-Curtis groupsArticle

Authors: Robert H. Gilman 1; Alexei G. Myasnikov 1

For any group $G$ and integer $k\ge 2$ the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group $AC_k(G)$, on the subset $N_k(G) \subset G^k$ of all $k$-tuples that generate $G$ as a normal subgroup (provided $N_k(G)$ is non-empty). The famous Andrews-Curtis Conjecture is that if $G$ is free of rank $k$, then $AC_k(G)$ acts transitively on $N_k(G)$. The set $N_k(G)$ may have a rather complex structure, so it is easier to study the full Andrews-Curtis group $FAC(G)$ generated by AC-transformations on a much simpler set $G^k$. Our goal here is to investigate the natural epimorphism $λ\colon FAC_k(G) \to AC_k(G)$. We show that if $G$ is non-elementary torsion-free hyperbolic, then $FAC_k(G)$ acts faithfully on every nontrivial orbit of $G^k$, hence $λ\colon FAC_k(G) \to AC_k(G)$ is an isomorphism.


Volume: Volume 16, Issue 1, Special issue in memory of Ben Fine
Published on: July 4, 2025
Accepted on: July 2, 2025
Submitted on: July 2, 2025
Keywords: Group Theory