Prosolvable rigidity of surface groups

• Andrei Jaikin-Zapirain ^[1] ; Ismael Morales ^[2]
  1. [1] Universidad Autónoma de Madrid

     Universidad Autónoma de Madrid

     Madrid, España

     
  2. [2] University of Oxford

     University of Oxford

     Oxford District, Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 5, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01082-1
• Enlaces
  • Texto completo
• Resumen

  • Surface groups are known to be the Poincaré duality groups of dimension two since the work of Eckmann, Linnell and Müller. We prove a prosolvable analogue of this result that allows us to show that surface groups are profinitely (and prosolvably) rigid among finitely generated groups that satisfy cd(G) = 2 and b(2) 2 (G) = 0. We explore two other consequences. On the one hand, we derive that if u is a surface word of a finitely generated free group F and v ∈ F is measure equivalent to u in all finite solvable quotients of F, then u and v belong to the same Aut(F)-orbit. Finally, we get a partial result towards Mel’nikov’s surface group conjecture. Let F be a free group of rank n 3 and let w ∈ F. Suppose that G = F/w is a residually finite group all of whose finite-index subgroups are one-relator groups. Then G is 2-free. Moreover, we show that if H2(G;Z) = 0 then G must be a surface group.

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