The finite and solvable genus of finitely generated free and surface groups
Let C be the pseudovariety F of all finite groups or the pseudovariety S of all finite solvable groups and let Γ be either a finitely generated free group or a surface group. The C -genus of Γ, denoted by GC(Γ), consists of the isomorphism classes of finitely generated residually- C groups G having...
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| Format: | article |
| Publication Date: | 2023 |
| Country: | España |
| Institution: | Universidad Autónoma de Madrid |
| Repository: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Language: | English |
| OAI Identifier: | oai:repositorio.uam.es:10486/709321 |
| Online Access: | http://hdl.handle.net/10486/709321 https://dx.doi.org/10.1007/s40687-023-00408-9 |
| Access Level: | Open access |
| Keyword: | Free Groups L -Betti Numbers 2 Profinite Rigidity Representation Varieties Surface Groups Matemáticas |
| Summary: | Let C be the pseudovariety F of all finite groups or the pseudovariety S of all finite solvable groups and let Γ be either a finitely generated free group or a surface group. The C -genus of Γ, denoted by GC(Γ), consists of the isomorphism classes of finitely generated residually- C groups G having the same quotients in C as Γ. We show that the groups from GC(Γ) are residually-p for all primes p. This answers a question of Gilbert Baumslag and shows that the groups in the genus are residually finite rationally solvable groups. This leads to a positive solution of particular case of a question of Alexander Grothendieck: if F is a free group, G is a finitely generated residually- C group and u: F→ G is a homomorphism such that the induced map of pro- C completions uC^: FC^→ GC^ is an isomorphism, then u is an isomorphism |
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