{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/97234"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/97234","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Compactness of the space of marked groups and examples of L2-Betti numbers of simple groups","abstract":"This paper contains two parts. The first part will introduce Gn space and will show its compact. I will give two proofs for the compactness, the first one is due to Rostislav Grigorchuk [1], which refers to geometrical group theory and after the first proof I will give a more topological proof. In the second part, our goal is to prove a theorem by Denis Osin and Andreas Thom [2]: for every integer n ≥ 2 and every ε ≥ 0 there exists an infinite simple group Q generated by n elements such that β(2)(Q) ≥ n − 1 − ε. As a corollary, we can prove that for every positive integer n 1 there exists a simple group Q with d(Q) = n. In the proof of this theorem, I added the details to the original proof. Moreover, I found and fixed an error of the original proof in [2], although it doesn’t affect the final result.","abstract_html":"This paper contains two parts. The first part will introduce Gn space and will show its compact. I will give two proofs for the compactness, the first one is due to Rostislav Grigorchuk [1], which refers to geometrical group theory and after the first proof I will give a more topological proof. In the second part, our goal is to prove a theorem by Denis Osin and Andreas Thom [2]: for every integer n ≥ 2 and every ε ≥ 0 there exists an infinite simple group Q generated by n elements such that β(2)(Q) ≥ n − 1 − ε. As a corollary, we can prove that for every positive integer n 1 there exists a simple group Q with d(Q) = n. In the proof of this theorem, I added the details to the original proof. Moreover, I found and fixed an error of the original proof in [2], although it doesn’t affect the final result.","abstract_has_math":false,"creators":["Zhu, Kejia"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Mineyev, Igor"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-08-10T19:14:13Z","date_published":"2017-08-10T19:14:13Z","updated_at":"2026-07-22T22:24:32Z","subjects":["Geometric group theory","Topology"],"languages":["en"],"rights":["Copyright 2017 Kejia Zhu"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/97234","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Mineyev, Igor"]},{"key":"dc:creator","label":"Author","values":["Zhu, Kejia"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-08-10T19:14:13Z","2017-03-28","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":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"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":["Geometric group theory","Topology"]}]},{"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 Kejia Zhu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/97234"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This paper contains two parts. The first part will introduce Gn space and will show its compact. I will give two proofs for the compactness, the first one is due to Rostislav Grigorchuk [1], which refers to geometrical group theory and after the first proof I will give a more topological proof. In the second part, our goal is to prove a theorem by Denis Osin and Andreas Thom [2]: for every integer n ≥ 2 and every ε ≥ 0 there exists an infinite simple group Q generated by n elements such that β(2)(Q) ≥ n − 1 − ε. As a corollary, we can prove that for every positive integer n 1 there exists a simple group Q with d(Q) = n. In the proof of this theorem, I added the details to the original proof. Moreover, I found and fixed an error of the original proof in [2], although it doesn’t affect the final result.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms","The student, Kejia Zhu, accepted the attached license on 2016-11-29 at 16:09.","The student, Kejia Zhu, submitted this Thesis for approval on 2017-03-27 at 15:14.","This Thesis was approved for publication on 2017-03-28 at 13:11.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10362 on 2017-08-10 at 13:36:29","Made available in DSpace on 2017-08-10T19:14:13Z (GMT). No. of bitstreams: 2 ZHU-THESIS-2017.pdf: 398597 bytes, checksum: 35355a9ea3978f6620bf2a166a14b135 (MD5) LICENSE.txt: 4206 bytes, checksum: c57ac2dec0caf041ca22dcbddcf02110 (MD5) Previous issue date: 2017-03-28"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Compactness of the space of marked groups and examples of L2-Betti numbers of simple groups"]}]}],"canonical_facts":{"dc:contributor":["Mineyev, Igor"],"dc:creator":["Zhu, Kejia"],"dc:date":["2017-08-10T19:14:13Z","2017-03-28","2017-05"],"dc:description":["This paper contains two parts. The first part will introduce Gn space and will show its compact. I will give two proofs for the compactness, the first one is due to Rostislav Grigorchuk [1], which refers to geometrical group theory and after the first proof I will give a more topological proof. In the second part, our goal is to prove a theorem by Denis Osin and Andreas Thom [2]: for every integer n ≥ 2 and every ε ≥ 0 there exists an infinite simple group Q generated by n elements such that β(2)(Q) ≥ n − 1 − ε. As a corollary, we can prove that for every positive integer n 1 there exists a simple group Q with d(Q) = n. In the proof of this theorem, I added the details to the original proof. Moreover, I found and fixed an error of the original proof in [2], although it doesn’t affect the final result.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms","The student, Kejia Zhu, accepted the attached license on 2016-11-29 at 16:09.","The student, Kejia Zhu, submitted this Thesis for approval on 2017-03-27 at 15:14.","This Thesis was approved for publication on 2017-03-28 at 13:11.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10362 on 2017-08-10 at 13:36:29","Made available in DSpace on 2017-08-10T19:14:13Z (GMT). No. of bitstreams: 2 ZHU-THESIS-2017.pdf: 398597 bytes, checksum: 35355a9ea3978f6620bf2a166a14b135 (MD5) LICENSE.txt: 4206 bytes, checksum: c57ac2dec0caf041ca22dcbddcf02110 (MD5) Previous issue date: 2017-03-28"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/97234"],"dc:language":["en"],"dc:rights":["Copyright 2017 Kejia Zhu"],"dc:subject":["Geometric group theory","Topology"],"dc:title":["Compactness of the space of marked groups and examples of L2-Betti numbers of simple groups"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["M.S."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:32Z"}