Andrews-Curtis groups

TL;DR

This paper investigates the structure of Andrews-Curtis groups acting on generating k-tuples of a group, proving that for certain hyperbolic groups, the simplified and full groups are isomorphic, shedding light on the Andrews-Curtis Conjecture.

Contribution

It establishes an isomorphism between the simplified and full Andrews-Curtis groups for non-elementary torsion-free hyperbolic groups.

Findings

FAC_k(G) acts faithfully on nontrivial orbits

The epimorphism λ is an isomorphism in this setting

Provides insight into the structure of Andrews-Curtis groups

Abstract

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 $\lambda\colon FAC_k(G) \to AC_k(G)$. 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 $\lambda\colon FAC_k(G) \to AC_k(G)$ is an isomorphism.