2026-09-19T05:05:43Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/97234 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Mineyev, Igor Zhu, Kejia 2017-08-10T19:14:13Z 2017-08-10T19:14:13Z 2017-03-28 2017-05 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 application/pdf http://hdl.handle.net/2142/97234 en Copyright 2017 Kejia Zhu Geometric group theory Topology Compactness of the space of marked groups and examples of L2-Betti numbers of simple groups text text Mathematics Mathematics University of Illinois at Urbana-Champaign Thesis M.S.