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Title:  Compactness of the space of marked groups and examples of L2Betti numbers of simple groups 
Author(s):  Zhu, Kejia 
Advisor(s):  Mineyev, Igor 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  M.S. 
Genre:  Thesis 
Subject(s):  Geometric group theory
Topology 
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. 
Issue Date:  20170328 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/97234 
Rights Information:  Copyright 2017 Kejia Zhu 
Date Available in IDEALS:  20170810 
Date Deposited:  201705 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois