The coarse geometry of group splittings
<p>This thesis addresses Gromov’s program of studying the geometry of finitely generated groups up to quasi-isometry. In particular, we investigate when we can characterise group splittings geometrically, and how this can be used to determine whether groups are quasi-isometric. </p>...
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | English |
| Published: |
2018
|
| Subjects: |
| _version_ | 1826285738218487808 |
|---|---|
| author | Margolis, A |
| author2 | Papazoglou, P |
| author_facet | Papazoglou, P Margolis, A |
| author_sort | Margolis, A |
| collection | OXFORD |
| description | <p>This thesis addresses Gromov’s program of studying the geometry of finitely generated groups up to quasi-isometry. In particular, we investigate when we can characterise group splittings geometrically, and how this can be used to determine whether groups are quasi-isometric. </p>
<p>In Chapters 2 and 3 we develop several coarse topological techniques, building on work of Kapovich–Kleiner [KK05], that we expect to be of interest in their own right. We use these techniques to show that if G is a group of type FP<sub>n+1</sub> that is coarsely 3-separated by an essentially embedded coarse PD<sub>n</sub> space W, then G splits over a subgroup H that is at finite Hausdorff distance from W. This can used to deduce that splittings of the form G = A <sub>*H</sub> B, where G is of type FP<sub>n+1</sub> and H is a coarse PD<sub>n</sub> group such that both |CommA(H) : H| and |CommB(H) : H| are greater than two, are invariant under quasi-isometry. These results are contained in [Mar18b]. </p>
<p>In Chapter 4 we extend the methods of Chapter 3 to understand the 2-separating case. Building on work of Scott–Swarup [SS03], we describe a <em>regular neighbourhood</em> of essentially embedded coarse PD<sub>n</sub> spaces. </p>
<p>In Chapter 5 we show that the JSJ tree of cylinders over abelian subgroups of a RAAG is a quasi-isometry invariant. This provides a necessary condition for two RAAGs to be quasi-isometric. We then restrict to the class of RAAGs whose JSJ trees of cylinders over infinite cyclic subgroups have abelian rigid vertex stabilizers. We show that two such RAAGs are quasi-isometric if and only if they have equivalent JSJ trees of cylinders. The latter result is a special case of results we prove in [Mar18a].</p>
|
| first_indexed | 2024-03-07T01:33:19Z |
| format | Thesis |
| id | oxford-uuid:944ea432-b2cd-48f3-9afe-df4bf96abbcd |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T01:33:19Z |
| publishDate | 2018 |
| record_format | dspace |
| spelling | oxford-uuid:944ea432-b2cd-48f3-9afe-df4bf96abbcd2022-03-26T23:38:32ZThe coarse geometry of group splittingsThesishttp://purl.org/coar/resource_type/c_db06uuid:944ea432-b2cd-48f3-9afe-df4bf96abbcdMathematicsEnglishORA Deposit2018Margolis, APapazoglou, P<p>This thesis addresses Gromov’s program of studying the geometry of finitely generated groups up to quasi-isometry. In particular, we investigate when we can characterise group splittings geometrically, and how this can be used to determine whether groups are quasi-isometric. </p> <p>In Chapters 2 and 3 we develop several coarse topological techniques, building on work of Kapovich–Kleiner [KK05], that we expect to be of interest in their own right. We use these techniques to show that if G is a group of type FP<sub>n+1</sub> that is coarsely 3-separated by an essentially embedded coarse PD<sub>n</sub> space W, then G splits over a subgroup H that is at finite Hausdorff distance from W. This can used to deduce that splittings of the form G = A <sub>*H</sub> B, where G is of type FP<sub>n+1</sub> and H is a coarse PD<sub>n</sub> group such that both |CommA(H) : H| and |CommB(H) : H| are greater than two, are invariant under quasi-isometry. These results are contained in [Mar18b]. </p> <p>In Chapter 4 we extend the methods of Chapter 3 to understand the 2-separating case. Building on work of Scott–Swarup [SS03], we describe a <em>regular neighbourhood</em> of essentially embedded coarse PD<sub>n</sub> spaces. </p> <p>In Chapter 5 we show that the JSJ tree of cylinders over abelian subgroups of a RAAG is a quasi-isometry invariant. This provides a necessary condition for two RAAGs to be quasi-isometric. We then restrict to the class of RAAGs whose JSJ trees of cylinders over infinite cyclic subgroups have abelian rigid vertex stabilizers. We show that two such RAAGs are quasi-isometric if and only if they have equivalent JSJ trees of cylinders. The latter result is a special case of results we prove in [Mar18a].</p> |
| spellingShingle | Mathematics Margolis, A The coarse geometry of group splittings |
| title | The coarse geometry of group splittings |
| title_full | The coarse geometry of group splittings |
| title_fullStr | The coarse geometry of group splittings |
| title_full_unstemmed | The coarse geometry of group splittings |
| title_short | The coarse geometry of group splittings |
| title_sort | coarse geometry of group splittings |
| topic | Mathematics |
| work_keys_str_mv | AT margolisa thecoarsegeometryofgroupsplittings AT margolisa coarsegeometryofgroupsplittings |