{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86819"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86819","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Quasiconformal Groups","abstract":"In this thesis the notion of Schottky quasiconformal groups is introduced. We study Schottky quasiconformal groups and show that the limit set L(G) of any Schottky quasi-conformal group G is uniformly perfect. Then the concept of Schottky-type quasiconformal groups is introduced, and we generalize our results about Schottky quasiconformal groups to the new setting. We prove that if G is a discrete non-elementary quasiconformal group then the Hausdorff dimension of the limit set L(G) is positive. We also prove that the general Cantor sets are uniformly perfect. We consider quasiconformal mappings, quasisymmetric mappings, and weakly quasiconformal mappings in metric spaces. We show that in Loewner spaces the fourth definition of quasiconformal mappings, i.e., the ring definition, is equivalent to the three principal definitions. We consider quasiconformal groups in Loewner spaces and generalize our results to Schottky quasiconformal groups in Loewner spaces.","abstract_html":"In this thesis the notion of Schottky quasiconformal groups is introduced. We study Schottky quasiconformal groups and show that the limit set L(G) of any Schottky quasi-conformal group G is uniformly perfect. Then the concept of Schottky-type quasiconformal groups is introduced, and we generalize our results about Schottky quasiconformal groups to the new setting. We prove that if G is a discrete non-elementary quasiconformal group then the Hausdorff dimension of the limit set L(G) is positive. We also prove that the general Cantor sets are uniformly perfect. We consider quasiconformal mappings, quasisymmetric mappings, and weakly quasiconformal mappings in metric spaces. We show that in Loewner spaces the fourth definition of quasiconformal mappings, i.e., the ring definition, is equivalent to the three principal definitions. We consider quasiconformal groups in Loewner spaces and generalize our results to Schottky quasiconformal groups in Loewner spaces.","abstract_has_math":false,"creators":["Li, Xiaosheng"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Aimo Hinkkanen"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:42Z","date_published":"2015-09-28T15:19:42Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3101901"],"render_values":[{"text":"(MiAaPQ)AAI3101901","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86819","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Aimo Hinkkanen"]},{"key":"dc:creator","label":"Author","values":["Li, Xiaosheng"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:42Z","10000-01-01","2003"]},{"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":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86819","(MiAaPQ)AAI3101901"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis the notion of Schottky quasiconformal groups is introduced. We study Schottky quasiconformal groups and show that the limit set L(G) of any Schottky quasi-conformal group G is uniformly perfect. Then the concept of Schottky-type quasiconformal groups is introduced, and we generalize our results about Schottky quasiconformal groups to the new setting. We prove that if G is a discrete non-elementary quasiconformal group then the Hausdorff dimension of the limit set L(G) is positive. We also prove that the general Cantor sets are uniformly perfect. We consider quasiconformal mappings, quasisymmetric mappings, and weakly quasiconformal mappings in metric spaces. We show that in Loewner spaces the fourth definition of quasiconformal mappings, i.e., the ring definition, is equivalent to the three principal definitions. We consider quasiconformal groups in Loewner spaces and generalize our results to Schottky quasiconformal groups in Loewner spaces.","Made available in DSpace on 2015-09-28T15:19:42Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3101901.pdf: 2337026 bytes, checksum: de6e0b2a045b7f288cf178c6b4b7fdff (MD5) Previous issue date: 2003","Embargo set by: Seth Robbins for item 88100 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","71 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2003."]},{"key":"dc:title","label":"Title","values":["Quasiconformal Groups"]}]}],"canonical_facts":{"dc:contributor":["Aimo Hinkkanen"],"dc:creator":["Li, Xiaosheng"],"dc:date":["2015-09-28T15:19:42Z","10000-01-01","2003"],"dc:description":["In this thesis the notion of Schottky quasiconformal groups is introduced. We study Schottky quasiconformal groups and show that the limit set L(G) of any Schottky quasi-conformal group G is uniformly perfect. Then the concept of Schottky-type quasiconformal groups is introduced, and we generalize our results about Schottky quasiconformal groups to the new setting. We prove that if G is a discrete non-elementary quasiconformal group then the Hausdorff dimension of the limit set L(G) is positive. We also prove that the general Cantor sets are uniformly perfect. We consider quasiconformal mappings, quasisymmetric mappings, and weakly quasiconformal mappings in metric spaces. We show that in Loewner spaces the fourth definition of quasiconformal mappings, i.e., the ring definition, is equivalent to the three principal definitions. We consider quasiconformal groups in Loewner spaces and generalize our results to Schottky quasiconformal groups in Loewner spaces.","Made available in DSpace on 2015-09-28T15:19:42Z (GMT). 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