Idea
If XX is a topological spaces, AA and BB to isomorphic subspaces of XX, and one is given a fundamental group of XX, what is the fundamental group of XX with attached handle from AA to BB ?
Definition
Given a group GG, and isomorphism ϕ:A→B\phi:A\to B of its subgroups, the Higman-Neumann-Neumann extension, or HNN extension is the quotient G⋆⟨t⟩/IG\star \langle t\rangle/I where II is the normal subgroup generated by t −1atϕ(t −1)t^{-1}a t\phi(t^{-1}) for all a∈Aa\in A.
Literature
Introduced in
- Graham Higman, B. H. Neumann, Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949) 247–254. MR32641
Standard reference
- Lyndon, Schupp, Combinatorial group theory
Recent survey
- arXiv:[2512.10800](https://arxiv.org/pdf/2512.108…
Idea
If XX is a topological spaces, AA and BB to isomorphic subspaces of XX, and one is given a fundamental group of XX, what is the fundamental group of XX with attached handle from AA to BB ?
Definition
Given a group GG, and isomorphism ϕ:A→B\phi:A\to B of its subgroups, the Higman-Neumann-Neumann extension, or HNN extension is the quotient G⋆⟨t⟩/IG\star \langle t\rangle/I where II is the normal subgroup generated by t −1atϕ(t −1)t^{-1}a t\phi(t^{-1}) for all a∈Aa\in A.
Literature
Introduced in
- Graham Higman, B. H. Neumann, Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949) 247–254. MR32641
Standard reference
- Lyndon, Schupp, Combinatorial group theory
Recent survey
- arXiv:2512.10800
Created on December 26, 2025 at 17:06:29. See the history of this page for a list of all contributions to it.