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In a Commentarii Mathematici Helvetici paper by Benno Eckman and Heinz Müller in 1980 (volume 50, pages 510-520) proved that poincaré Duality Groups of dimension 2 with positive first Betti number are surface groups.

Is there any development into proving that certain groups of (homological) or geometric dimension 2 are surface groups?

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    $\begingroup$ The term is "homological dimension 2", where "homological" qualifies "dimension". So "homological 2 dimensional" and "homologic dimension 2" both sound weird. $\endgroup$
    – YCor
    Commented Jan 6, 2016 at 22:44
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    $\begingroup$ It is very unclear what exactly your question is. It is known that PD(2) groups over Z are surface groups; it is also known that PD(2) groups over other "reasonable" commutative rings are virtually surface groups. No need to assume nontriviality of the 1st betti number. What are the "certain" groups you are interested in? $\endgroup$
    – Misha
    Commented Jan 7, 2016 at 7:52
  • $\begingroup$ Dear Misha, I would like to know a reference for the result without the 1st Betti number assumption, both over $\mathbb{Z}$ and for the "virtual" statement for "reasonable" commutative rings. Thanks a lot in advance! $\endgroup$
    – Nicolas Boerger
    Commented Jan 7, 2016 at 22:27

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See my paper with Bruce Kleiner "Geometry of quasiplanes" for general unital commutative rings, it also contains a reference to the paper by Eckmann and Linnell from 1983 where they prove the theorem for PD(2) groups over Z and to the one by Bowditch which works over Q.

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