GAFA Geometric And Functional Analysis

, Volume 17, Issue 2, pp 385-403

Subgroups Of Direct Products Of Elementarily Free Groups

  • Martin R. BridsonAffiliated withDepartment of Mathematics, Imperial College Email author 
  • , James HowieAffiliated withMaxwell Institute of Mathematical Sciences, Heriot–Watt University

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The structure of groups having the same elementary theory as free groups is now known: they and their finitely generated subgroups form a prescribed subclass \({\mathcal{E}}\) of the hyperbolic limit groups. We prove that if G 1,...,G n are in \({\mathcal{E}}\) then a subgroup Γ ⊂ G 1 × … × G n is of type FP n if and only if Γ is itself, up to finite index, the direct product of at most n groups from \({\mathcal{E}}\) . This provides a partial answer to a question of Sela.

Keywords and phrases:

Limit groups homological finiteness properties Bass–Serre theory

AMS Mathematics Subject Classification:

20F65 20E08 20F67