{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/107885"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/107885","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Trees, dendrites, and the Cannon-Thurston map","abstract":"Made available in DSpace on 2020-08-26T21:54:20Z (GMT). No. of bitstreams: 7 FIELD-DISSERTATION-2020.pdf: 526393 bytes, checksum: 55c9b0a2e5c08967f5151d583f4ea47c (MD5) AlgebraicLamination.pdf: 11632 bytes, checksum: b93bb9ed7222c1b733ecf6d7808ed116 (MD5) EFieldThesis_Revised.tex: 200721 bytes, checksum: 8a6efacd69995a9d8f6ad410113b6cd2 (MD5) bib.bib: 21110 bytes, checksum: 3e76abdb66b898f27d39b394855bdf6f (MD5) uiucthesis2014.cls: 17030 bytes, checksum: 0c226f13da5d7c98fed1171999039c62 (MD5) uiucthesis2014.sty: 16943 bytes, checksum: c1c722f1968c3391a08a5058870fb61e (MD5) LICENSE.txt: 4212 bytes, checksum: 8245df692e78044c88edf7bfcc9db213 (MD5) Previous issue date: 2020-04-16","abstract_html":"Made available in DSpace on 2020-08-26T21:54:20Z (GMT). No. of bitstreams: 7 FIELD-DISSERTATION-2020.pdf: 526393 bytes, checksum: 55c9b0a2e5c08967f5151d583f4ea47c (MD5) AlgebraicLamination.pdf: 11632 bytes, checksum: b93bb9ed7222c1b733ecf6d7808ed116 (MD5) EFieldThesis_Revised.tex: 200721 bytes, checksum: 8a6efacd69995a9d8f6ad410113b6cd2 (MD5) bib.bib: 21110 bytes, checksum: 3e76abdb66b898f27d39b394855bdf6f (MD5) uiucthesis2014.cls: 17030 bytes, checksum: 0c226f13da5d7c98fed1171999039c62 (MD5) uiucthesis2014.sty: 16943 bytes, checksum: c1c722f1968c3391a08a5058870fb61e (MD5) LICENSE.txt: 4212 bytes, checksum: 8245df692e78044c88edf7bfcc9db213 (MD5) Previous issue date: 2020-04-16","abstract_has_math":false,"creators":["Field, Elizabeth C"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Kapovich, Ilya","Leininger, Christopher J","Dunfield, Nathan","Schupp, Paul"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-08-26T21:54:20Z","date_published":"2020-08-26T21:54:20Z","updated_at":"2026-07-22T22:24:47Z","subjects":["Cannon-Thurston map","hyperbolic group","algebraic lamination","dendrite","Gromov boundary"],"languages":["en"],"rights":["Copyright 2020 Elizabeth Field"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/107885","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kapovich, Ilya","Leininger, Christopher J","Dunfield, Nathan","Schupp, Paul"]},{"key":"dc:creator","label":"Author","values":["Field, Elizabeth C"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-08-26T21:54:20Z","2020-04-16","2020-05"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"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":["Cannon-Thurston map","hyperbolic group","algebraic lamination","dendrite","Gromov boundary"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2020 Elizabeth Field"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/107885"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Made available in DSpace on 2020-08-26T21:54:20Z (GMT). No. of bitstreams: 7 FIELD-DISSERTATION-2020.pdf: 526393 bytes, checksum: 55c9b0a2e5c08967f5151d583f4ea47c (MD5) AlgebraicLamination.pdf: 11632 bytes, checksum: b93bb9ed7222c1b733ecf6d7808ed116 (MD5) EFieldThesis_Revised.tex: 200721 bytes, checksum: 8a6efacd69995a9d8f6ad410113b6cd2 (MD5) bib.bib: 21110 bytes, checksum: 3e76abdb66b898f27d39b394855bdf6f (MD5) uiucthesis2014.cls: 17030 bytes, checksum: 0c226f13da5d7c98fed1171999039c62 (MD5) uiucthesis2014.sty: 16943 bytes, checksum: c1c722f1968c3391a08a5058870fb61e (MD5) LICENSE.txt: 4212 bytes, checksum: 8245df692e78044c88edf7bfcc9db213 (MD5) Previous issue date: 2020-04-16","When $1\\to H\\to G\\to Q\\to 1$ is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from $H$ to $G$ extends continuously to a map between the Gromov boundaries of $H$ and $G$. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point $z$ in the Gromov boundary of $Q$ an ``ending lamination'' on $H$ which consists of pairs of distinct points in the boundary of $H$. We prove that for each such $z$, the quotient of the Gromov boundary of $H$ by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where $H$ is a free group and $Q$ is a convex cocompact purely atoroidal subgroup of $\\mathrm{Out}(F_N)$, one can identify the resultant quotient space with a certain $\\mathbb{R}$-tree in the boundary of Culler-Vogtmann's Outer space.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Elizabeth Field, accepted the attached license on 2020-04-15 at 09:26.","The student, Elizabeth Field, submitted this Dissertation for approval on 2020-04-15 at 09:37.","This Dissertation was approved for publication on 2020-04-16 at 11:13.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14981 on 2020-08-25 at 17:07:01"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Trees, dendrites, and the Cannon-Thurston map"]}]}],"canonical_facts":{"dc:contributor":["Kapovich, Ilya","Leininger, Christopher J","Dunfield, Nathan","Schupp, Paul"],"dc:creator":["Field, Elizabeth C"],"dc:date":["2020-08-26T21:54:20Z","2020-04-16","2020-05"],"dc:description":["Made available in DSpace on 2020-08-26T21:54:20Z (GMT). No. of bitstreams: 7 FIELD-DISSERTATION-2020.pdf: 526393 bytes, checksum: 55c9b0a2e5c08967f5151d583f4ea47c (MD5) AlgebraicLamination.pdf: 11632 bytes, checksum: b93bb9ed7222c1b733ecf6d7808ed116 (MD5) EFieldThesis_Revised.tex: 200721 bytes, checksum: 8a6efacd69995a9d8f6ad410113b6cd2 (MD5) bib.bib: 21110 bytes, checksum: 3e76abdb66b898f27d39b394855bdf6f (MD5) uiucthesis2014.cls: 17030 bytes, checksum: 0c226f13da5d7c98fed1171999039c62 (MD5) uiucthesis2014.sty: 16943 bytes, checksum: c1c722f1968c3391a08a5058870fb61e (MD5) LICENSE.txt: 4212 bytes, checksum: 8245df692e78044c88edf7bfcc9db213 (MD5) Previous issue date: 2020-04-16","When $1\\to H\\to G\\to Q\\to 1$ is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from $H$ to $G$ extends continuously to a map between the Gromov boundaries of $H$ and $G$. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point $z$ in the Gromov boundary of $Q$ an ``ending lamination'' on $H$ which consists of pairs of distinct points in the boundary of $H$. We prove that for each such $z$, the quotient of the Gromov boundary of $H$ by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where $H$ is a free group and $Q$ is a convex cocompact purely atoroidal subgroup of $\\mathrm{Out}(F_N)$, one can identify the resultant quotient space with a certain $\\mathbb{R}$-tree in the boundary of Culler-Vogtmann's Outer space.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Elizabeth Field, accepted the attached license on 2020-04-15 at 09:26.","The student, Elizabeth Field, submitted this Dissertation for approval on 2020-04-15 at 09:37.","This Dissertation was approved for publication on 2020-04-16 at 11:13.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14981 on 2020-08-25 at 17:07:01"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/107885"],"dc:language":["en"],"dc:rights":["Copyright 2020 Elizabeth Field"],"dc:subject":["Cannon-Thurston map","hyperbolic group","algebraic lamination","dendrite","Gromov boundary"],"dc:title":["Trees, dendrites, and the Cannon-Thurston map"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:47Z"}