• Boundaries
• Section7

7 Asymptotic topology

Problems below are mostly motivated by the following rigidity results of Panos Papasoglu, MR2153400:

Theorem 7.1 Suppose that

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is a finitely-presented 1-ended group. Then:

1. The JSJ decomposition of

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   is invariant under quasi-isometries.

2. A quasiline coarsely separates Cayley graph of

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   iff

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   splits over virtually-

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   or

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   is virtually a surface group.

3. No quasi-ray coarsely separates the Cayley graph of

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   .

Problem 7.1 (Comments) [Panos Papasoglu] Do these results hold for general finitely generated groups?

Problem 7.2 (Comments) [Panos Papasoglu] Are splittings over

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(or

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) invariant under quasi-isometry? The analogous problem also makes sense for the JSJ decompositions.

Problem 7.3 (Comments) [Panos Papasoglu] Suppose

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is finitely generated and there is a sequence of quasicircles that separate its Cayley graph. Is

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virtually a surface group?

Problem 7.4 (Comments) [Conjecture of Panos Papasoglu] If

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is finitely generated with asymptotic dimension

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, and

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is a subset of the Cayley graph with asymptotic dimension

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that coarsely separates the Cayley graph, then

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splits over some subgroup

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with asymptotic dimension

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.

A homogeneous continuum is a locally connected compact metric space whose group of homeomorphisms acts transitively. Papasoglu showed that every simply connected homogeneous continuum has the property that no simple arc separates it.

Problem 7.5 (Comments) [Panos Papasoglu] Do all homogeneous continua (with dimension greater than 2) have this property?

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