{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/374321"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/374321","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Computations and lower bounds for scl and the relative Gromov seminorm","abstract":"The main purpose of this thesis is to give computations and estimates of stable commutator length, an invariant of groups that measures homological complexity and is connected to several notions of negative curvature in geometric group theory. A first aspect of this is to give exact computations, motivated by questions of rationality. A particular case of interest is that of surface groups. We introduce the relative Gromov seminorm, a new invariant that serves as an intermediate step in computations of scl. We show that several results about scl in free groups generalise to the relative Gromov seminorm in surface groups. We explain how bounded cohomology provides a dual to the relative Gromov seminorm, and apply those ideas to obtain new computations of scl. Another aspect of this thesis is to obtain lower bounds for stable commutator length in the presence of negative curvature. We do this by developing a new geometric method, where surfaces estimating scl are equipped with a combinatorial geometric structure called an angle structure, and for which there is a notion of curvature and a version of the Gauß–Bonnet formula leading to estimates of the Euler characteristic. 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Another aspect of this thesis is to obtain lower bounds for stable commutator length in the presence of negative curvature. We do this by developing a new geometric method, where surfaces estimating scl are equipped with a combinatorial geometric structure called an angle structure, and for which there is a notion of curvature and a version of the Gauß–Bonnet formula leading to estimates of the Euler characteristic. 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