Andrews-Curtis groups

2025/06/28 by Gilman, Robert H., Myasnikov, Alexei G.
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2506.23031

Abstract

For any group G and integer k≥ 2 the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group AC_k(G), on the subset N_k(G) ⊂ 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) → 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) → AC_k(G) is an isomorphism.

Citations

• What makes math problems hard for reinforcement learning: a case study
• Conjugacy search problem and the Andrews-Curtis conjecture
• The complexity of balanced presentations and the Andrews-Curtis conjecture
• On the Andrews-Curtis equivalence
• The Andrews-Curtis Conjecture and Black Box Groups
• Genetic algorithms and the Andrews-Curtis conjecture
• CSA groups and separated free constructions
• Exponential groups 2: Extensions of centralizers and tensor completion of CSA groups

Related

• On the Andrews-Curtis groups: non-finite presentability
• The Andrews-Curtis Conjecture and Black Box Groups
• On Andrews-Curtis conjectures for soluble groups
• On the Andrews-Curtis equivalence
• On conjectures of Andrews and Curtis