{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/97398"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/97398","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Gromov boundaries of complexes associated to surfaces","abstract":"In 1996, Masur and Minsky showed that the curve graph is hyperbolic. Recently, Hensel, Przytycki, and Webb proved a stronger result which was the uniform hyperbolicity of the curve graph, and they also gave the first proof of the uniform hyperbolicity of the arc graph using unicorn arcs. For closed surfaces, their proof is indirect, but Przytycki and Sisto gave a more direct proof of hyperbolicity in that case using bicorn curves. In this dissertation, we extend the notion of unicorn arcs and bicorn curves between two arcs or curves to the case where we replace one arc or curve with a geodesic asymptotic to a lamination or a leaf of the lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the arc graph, respectively, as spaces of laminations.","abstract_html":"In 1996, Masur and Minsky showed that the curve graph is hyperbolic. Recently, Hensel, Przytycki, and Webb proved a stronger result which was the uniform hyperbolicity of the curve graph, and they also gave the first proof of the uniform hyperbolicity of the arc graph using unicorn arcs. For closed surfaces, their proof is indirect, but Przytycki and Sisto gave a more direct proof of hyperbolicity in that case using bicorn curves. In this dissertation, we extend the notion of unicorn arcs and bicorn curves between two arcs or curves to the case where we replace one arc or curve with a geodesic asymptotic to a lamination or a leaf of the lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the arc graph, respectively, as spaces of laminations.","abstract_has_math":false,"creators":["Pho-on, Witsarut"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Leininger, Christopher","Dunfield, Nathan","Kapovich, Ilya","Bradlow, Steven"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-08-10T19:15:24Z","date_published":"2017-08-10T19:15:24Z","updated_at":"2026-07-22T22:24:34Z","subjects":["Gromov boundary","Curve complex","Arc complex","Lamination","Surface","Unicorn curve","Bicorn curve"],"languages":["en"],"rights":["Copyright 2017 Witsarut Pho-on"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/97398","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Leininger, Christopher","Dunfield, Nathan","Kapovich, Ilya","Bradlow, Steven"]},{"key":"dc:creator","label":"Author","values":["Pho-on, Witsarut"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-08-10T19:15:24Z","2017-04-19","2017-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Gromov boundary","Curve complex","Arc complex","Lamination","Surface","Unicorn curve","Bicorn curve"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Witsarut Pho-on"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/97398"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In 1996, Masur and Minsky showed that the curve graph is hyperbolic. Recently, Hensel, Przytycki, and Webb proved a stronger result which was the uniform hyperbolicity of the curve graph, and they also gave the first proof of the uniform hyperbolicity of the arc graph using unicorn arcs. For closed surfaces, their proof is indirect, but Przytycki and Sisto gave a more direct proof of hyperbolicity in that case using bicorn curves. In this dissertation, we extend the notion of unicorn arcs and bicorn curves between two arcs or curves to the case where we replace one arc or curve with a geodesic asymptotic to a lamination or a leaf of the lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the arc graph, respectively, as spaces of laminations.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms","The student, Witsarut Pho-on, accepted the attached license on 2017-04-18 at 16:25.","The student, Witsarut Pho-on, submitted this Dissertation for approval on 2017-04-18 at 16:58.","This Dissertation was approved for publication on 2017-04-19 at 16:07.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10866 on 2017-08-10 at 13:42:09","Made available in DSpace on 2017-08-10T19:15:24Z (GMT). No. of bitstreams: 2 PHO-ON-DISSERTATION-2017.pdf: 484958 bytes, checksum: acc8945bad63dfc7a3cdd68960635db4 (MD5) LICENSE.txt: 4212 bytes, checksum: 274b28e37dd776bb8fa80d161ecf1df6 (MD5) Previous issue date: 2017-04-19"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Gromov boundaries of complexes associated to surfaces"]}]}],"canonical_facts":{"dc:contributor":["Leininger, Christopher","Dunfield, Nathan","Kapovich, Ilya","Bradlow, Steven"],"dc:creator":["Pho-on, Witsarut"],"dc:date":["2017-08-10T19:15:24Z","2017-04-19","2017-05"],"dc:description":["In 1996, Masur and Minsky showed that the curve graph is hyperbolic. Recently, Hensel, Przytycki, and Webb proved a stronger result which was the uniform hyperbolicity of the curve graph, and they also gave the first proof of the uniform hyperbolicity of the arc graph using unicorn arcs. For closed surfaces, their proof is indirect, but Przytycki and Sisto gave a more direct proof of hyperbolicity in that case using bicorn curves. In this dissertation, we extend the notion of unicorn arcs and bicorn curves between two arcs or curves to the case where we replace one arc or curve with a geodesic asymptotic to a lamination or a leaf of the lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the arc graph, respectively, as spaces of laminations.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms","The student, Witsarut Pho-on, accepted the attached license on 2017-04-18 at 16:25.","The student, Witsarut Pho-on, submitted this Dissertation for approval on 2017-04-18 at 16:58.","This Dissertation was approved for publication on 2017-04-19 at 16:07.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10866 on 2017-08-10 at 13:42:09","Made available in DSpace on 2017-08-10T19:15:24Z (GMT). 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