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Title: The coarse geometry of group splittings
Author: Margolis, Alex
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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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. 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 FPn+1 that is coarsely 3-separated by an essentially embedded coarse PDn 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 *H B, where G is of type FPn+1 and H is a coarse PDn 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]. 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 regular neighbourhood of essentially embedded coarse PDn spaces. 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].
Supervisor: Papazoglou, Panagiotis Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics