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2. Accessibility Questions

We recall the definition of strong accessibility of a group G over a family of subgroups \mathcal C: decompose G as a graph of groups with edge groups in \mathcal C. Then decompose the vertex groups of this graph as graphs of groups with edge groups in \mathcal C and so on. If there is such a series of decompositions that terminates (i.e. at some stage the vertex groups do not admit any decomposition over \mathcal C) then the group is called strongly accessible over \mathcal C. We note that one asks only for the existence of such a series of decompositions and not that every such series terminates. This is analogous to hierarchies for 3-manifolds.

    1. Add remark

      Problem 2.1.

      [Swarup] Let {\mathcal C} be the class of finite and virtually-\mathbb Z groups. Are hyperbolic groups strongly accessible over {\mathcal C}?
        1. Remark. Grushko’s theorem implies that finitely generated groups are strongly accessible over the trivial group. Dunwoody’s theorem [MR807066] implies that finitely presented groups are strongly accessible over {\mathcal C}=\{\text {finite groups}\}. Delzant and Potyagailo [MR1838998] showed that hyperbolic groups with no 2 torsion are strongly accessible over the class {\mathcal C} of finite and 2-ended groups. So the question remaining here is whether the no-2 torsion assumption can be removed.

            • Add remark

              Problem 2.2.

              [Swarup] Let {\mathcal C} be the class of finite and virtually-\mathbb Z groups. Are finitely presented groups strongly accessible over {\mathcal C}?
                1. Remark. One can ask this more generally for splittings over small groups.

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                      Problem 2.3.

                      [Sageev] Is there a 1-ended finitely presented group G such that G\simeq G*_{\mathbb Z}?

                        •     One may ask more generally

                          Add remark

                          Problem 2.4.

                          Let {\mathcal C} be the class of finitely generated groups. Are finitely presented groups strongly accessible over {\mathcal C}?

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                              Problem 2.5.

                              Let G=A*_{C_n}B_n,\, n\in \mathbb N where G is a finitely generated, C_n are slender and C_n < C_{n+1} for all n. Is it true that G=A*_C B where C < C_n for all n? Similarly for HNN extensions.
                                1. Remark. This was shown by Rips-Sela [MR1469317] for C_n isomorphic to \mathbb Z. Weidmann pointed out that one may pose the question even more generally omitting ‘finitely generated’ for G. It is true for finitely presented groups [MR2221253].

                                      Cite this as: AimPL: JSJ Decompositions, available at http://aimpl.org/jsjdecomp.