On finitely generated profinite groups, I: strong completeness and uniform bounds

By Nikolay Nikolov, D. Segal. 2006

273 citations recorded by the source corpus.

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let w be a 'locally finite' group word and d ∈ N. Then there exists f = f(w, d) such that in every d-generator finite group G, every element of the verbal subgroup w(G) is equal to a product of f w-values. An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed. The proofs rely on some properties of the finite simple groups, to be established in Part II.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b473506b16129e5842483e824dbfc458e732fb43

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In Algebra and Number Theory

1. Alan W. Reid
2. Martin R. Bridson
3. Marston D. E. Conder
4. Gareth Wilkes

In Discrete Mathematics and Combinatorics

1. Gareth Wilkes
2. Alan W. Reid
3. Martin R. Bridson
4. Marston D. E. Conder

In Geometry and Topology

1. Gareth Wilkes
2. Alan W. Reid
3. Martin R. Bridson
4. Marston D. E. Conder

Citations in the corpus

• Word maps and Waring type problems (2007, unknown): “A recent deep result of Nikolov and Segal [NS1, NS2] shows that this holds for every finitely generated profinite group.”
• Characters of symmetric groups: sharp bounds and applications (2008, unknown): “See Borel [Bo] for algebraic groups, Nikolov and Segal [ NS1 ], [NS2] for profinite groups, as well as [LiSh1], [DPSSh], [L], [Sh] and [LaSh].”
• Determining Fuchsian groups by their finite quotients (2014, methodology): “Following the aforementioned work of Nikolov and Segal [44], by ‘isomorphic’ we simply mean ‘isomorphic as groups’, for by [44], each group-theoretic isomorphism is continuous, and so any such isomorphism will be an isomorphism of topological groups.”
• Profinite rigidity for Seifert fibre spaces (2015, methodology): “The open subgroups of Ĝ are in fact all its finite index subgroups, by the Nikolov-Segal theorem [NS07].”