3 Accessibility Questions
We recall the definition of strong accessibility of a group
dvipng error! exitcode was 2 (signal 0), transscript follows:over a family of subgroups
dvipng error! exitcode was 2 (signal 0), transscript follows:: decompose
dvipng error! exitcode was 2 (signal 0), transscript follows:as a graph of groups with edge groups in
dvipng error! exitcode was 2 (signal 0), transscript follows:. Then decompose the vertex groups of this graph as graphs of groups with edge groups in
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:) then the group is called strongly accessible over
dvipng error! exitcode was 2 (signal 0), transscript follows:. 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.
Problem 3.1 (Comments) (Swarup) Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the class of finite and virtually-
dvipng error! exitcode was 2 (signal 0), transscript follows:groups. Are hyperbolic groups strongly accessible over
dvipng error! exitcode was 2 (signal 0), transscript follows:?
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
dvipng error! exitcode was 2 (signal 0), transscript follows:. Delzant and Potyagailo MR1838998 showed that hyperbolic groups with no 2 torsion are strongly accessible over the class
dvipng error! exitcode was 2 (signal 0), transscript follows:of finite and 2-ended groups. So the question remaining here is whether the no-2 torsion assumption can be removed.
Problem 3.2 (Comments) (Swarup) Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the class of finite and virtually-
dvipng error! exitcode was 2 (signal 0), transscript follows:groups. Are finitely presented groups strongly accessible over
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Remark One can ask this more generally for splittings over small groups.
Problem 3.3 (Comments) (Sageev) Is there a 1-ended finitely presented group
dvipng error! exitcode was 2 (signal 0), transscript follows:such that
dvipng error! exitcode was 2 (signal 0), transscript follows:?
One may ask more generally
Problem 3.4 (Comments) Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the class of finitely generated groups. Are finitely presented groups strongly accessible over
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Problem 3.5 (Comments) Let
dvipng error! exitcode was 2 (signal 0), transscript follows:where
dvipng error! exitcode was 2 (signal 0), transscript follows:is a finitely generated,
dvipng error! exitcode was 2 (signal 0), transscript follows:are slender and
dvipng error! exitcode was 2 (signal 0), transscript follows:for all
dvipng error! exitcode was 2 (signal 0), transscript follows:. Is it true that
dvipng error! exitcode was 2 (signal 0), transscript follows:where
dvipng error! exitcode was 2 (signal 0), transscript follows:for all
dvipng error! exitcode was 2 (signal 0), transscript follows:? Similarly for HNN extensions.
Remark This was shown by Rips-Sela MR1469317 for
dvipng error! exitcode was 2 (signal 0), transscript follows:isomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:. Weidmann pointed out that one may pose the question even more generally omitting `finitely generated' for
dvipng error! exitcode was 2 (signal 0), transscript follows:. It is true for finitely presented groups MR2221253.