Group splittings and asymptotic topology · Question 3.8
JSJ Decompositions · AIMPL workshop record
Editorial mapping of AIMPL source category: Geometric group theory. http://aimpl.org/jsjdecomp/ Not an author-supplied MSC code.
Problem statement
Let be a locally connected homogeneous continuum of dimension . Is it true that is not separated by an arc?
Lean 4
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Background
Workshop introduction
Splitting theory of groups parallels decomposition theorems in 3-manifold theory. Kneser's theorem has its analog in Grushko's theorem, Haken finiteness and hierarchies gave rise to accessibility and strong accessibility in group theory while JSJ decompositions of 3-manifolds were shown recently to extend to group theory too. One should stress that these "generalizations" from 3-manifolds to groups were by no means routine and a lot of questions remain open today. The structure theory that one obtains for groups is often much more general that the 3-manifold theory and it had many striking applications (isomorphism problem for hyperbolic groups, Tarski problems etc). We refer to Source citation: MR1469317 and Source citation: MR1664694 for the basic definitions.
Section context
One may pose a very general question: What meaningful geometric properties distinguish Cayley graphs from arbitrary metric spaces? Here geometric of course is in the sense of Gromov's quasi-isometries. A lot of the recent work in geometric group theory falls in this frame. Stallings' ends theorem, Gromov's polynomial growth theorem and Varopoulos isoperimetric inequality are three deep positive results distinguishing Cayley graphs from general metric spaces. On the other hand one may see Grigorchuk's intermediate growth examples and Rips-Sapir-Birget-Olshanskii-Bridson isoperimetric inequalities examples as negative results to the above question, saying that "everything is possible" for Cayley graphs.
Splitting theory has provided some further positive results in this quest and there is some reasonable hope for further progress. We state here some specific questions in this spirit.
Question context
It is interesting to note that some of the above questions can be stated also for locally connected continua (especially homogeneous continua). For example the question on quasi-rays that separate has the following twin:
Progress & source assessment
Dated source record, inspected 2026-09-10. Current status has not been independently reassessed. Source-reported results have not been independently verified.
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- [1]JSJ Decompositions · Original question link
- [2]AIMPL source bibliography link
- [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.
Recorded progress
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