Timeline for Classification of groups in which the centralizer of every non-identity element is cyclic

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Nov 19, 2023 at 19:02	comment	added	Lvzhou Chen		Besides, by the recent work on hyperbolic 5-manifolds that fibers over S^1 by Italiano-Martelli-Migliorini, there are torsion-free hyperbolic groups that contain a subgroup H of finite type but not hyperbolic. Such a group H gives a stronger answer than Brady's example.
Nov 19, 2023 at 18:43	comment	added	Lvzhou Chen		Thank you, Henry, for writing down more details! Just to add a reference. Brady's example is in the paper titled "Branched Coverings of Cubical Complexes and Subgroups of Hyperbolic Groups".
Apr 13, 2017 at 12:58	history	edited	CommunityBot		replaced http://mathoverflow.net/ with https://mathoverflow.net/
Jun 26, 2015 at 8:54	comment	added	HJRW		PS Ian, I know you know this. But I thought it worth setting the record straight.
Jun 26, 2015 at 8:53	comment	added	HJRW		In fact, Rips found (torsion-free) examples of finitely generated but infinitely presented (and hence not hyperbolic) subgroups of hyperbolic groups in the early '80s. I think Gromov may have asked whether every such finitely presented group is word-hyperbolic. A (torsion-free) counterexample was found by Noel Brady. I think the correct statement is now 'For torsion-free group of type $F_3$, your question is as difficult as Gromov's question.'
Jan 26, 2014 at 2:14	comment	added	HJRW		Good point!{}{}
Jan 26, 2014 at 0:10	comment	added	Ian Agol		@HJRW: it contains $\mathbb{Z}^2$. In fact, it can be regarded as an HNN extension of the trefoil knot group, identifying the cyclic subgroups of the 2 exceptional fibers.
Jan 25, 2014 at 23:20	comment	added	HJRW		Doesn't $BS(2,3)$ contain no solvable Baumslag--Solitar subgroup?
Jan 25, 2014 at 19:35	history	answered	Ian Agol	CC BY-SA 3.0