Year: 2025
Type: article
Abstract: 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 $${{\,\textrm{cd}\,}}(G)=2$$ cd ( G ) = 2 and $$b_2^{(2)}(G)=0$$ 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 $$v\in F$$ v ∈ F is measure equivalent to u in all finite solvable quotients of F , then u and v belong to the same $${{\,\textrm{Aut}\,}}(F)$$ Aut ( F ) -orbit. Finally, we get a partial result towards Mel’nikov’s surface group conjecture. Let F be a free group of rank $$n\geqslant 3$$ n ⩾ 3 and let $$w\in F$$ w ∈ F . Suppose that $$G=F/\langle \!\langle w\rangle \!\rangle $$ 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 $$H^2(G; \mathbb {Z})\ne 0$$ H 2 ( G ; Z ) ≠ 0 then G must be a surface group.
Source: Selecta Mathematica
Authors: Andrei Jaikin-Zapirain, Ismael Morales
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4414339520