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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 application/pdf Kapovich, Ilya Leininger, Christopher J Dunfield, Nathan Schupp, Paul Field, Elizabeth C 2020-08-26T21:54:20Z 2020-08-26T21:54:20Z 2020-04-16 2020-05 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 http://hdl.handle.net/2142/107885 en Copyright 2020 Elizabeth Field Cannon-Thurston map hyperbolic group algebraic lamination dendrite Gromov boundary Trees, dendrites, and the Cannon-Thurston map text Thesis Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.