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Prosolvable rigidity of surface groups

Andrei Jaikin-Zapirain, Ismael Morales

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 $\mathrm{cd}(G)=2$ 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 $v\in F$ is measure equivalent to $u$ in all finite solvable quotients of $F$ then $u$ and $v$ belong to the same $\mathrm{Aut}(F)$-orbit. Finally, we get a partial result towards Mel'nikov's surface group conjecture. Let $F$ be a free group of rank $n\geq 3$ and let $w\in F$. Suppose that $G=F/\langle\!\langle w\rangle\!\rangle$ 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 $H^2(G; \mathbb{Z})\neq 0$ then $G$ must be a surface group.

Prosolvable rigidity of surface groups

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 and . We explore two other consequences. On the one hand, we derive that if is a surface word of a finitely generated free group and is measure equivalent to in all finite solvable quotients of then and belong to the same -orbit. Finally, we get a partial result towards Mel'nikov's surface group conjecture. Let be a free group of rank and let . Suppose that is a residually finite group all of whose finite-index subgroups are one-relator groups. Then is 2-free. Moreover, we show that if then must be a surface group.
Paper Structure (17 sections, 22 theorems, 38 equations)

This paper contains 17 sections, 22 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Profinite rigidity.
  3. Measure equivalence.
  4. The surface group conjecture.
  5. More details on our methods
  6. Preliminaries
  7. The Euler characteristic
  8. Finiteness properties
  9. Pseudo-varieties of finite groups and profinite completions
  10. $L^2$-Betti numbers and their mod-$p$ variants for RFRS groups
  11. Poincaré duality groups of dimension 2
  12. Cohomology of profinite groups
  13. Cohomological goodness
  14. The proof of Theorem \ref{['main0']}
  15. The proof of Theorem \ref{['main']}
  16. ...and 2 more sections

Key Result

Theorem A

Let $p$ be a prime and let $G$ be a RFRS group of cohomological dimension two and type $\mathtt{FP}_2(\mathbb{F}_p)$. Suppose that $G$ is prosolvable $p$-good and that its prosolvable completion $G_{\widehat{\mathcal{S}}}$ is Poincaré Duality of dimension two at $p$. Then $G$ is a surface group.

Theorems & Definitions (80)

  • Theorem A
  • Theorem B
  • Conjecture 1
  • Theorem C
  • Conjecture 2: Mel'nikov
  • Theorem D
  • Remark
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 70 more