Compactness of the space of marked groups and examples of L2-Betti numbers of simple groups
Zhu, Kejia
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https://hdl.handle.net/2142/97234
Description
Title
Compactness of the space of marked groups and examples of L2-Betti numbers of simple groups
Author(s)
Zhu, Kejia
Issue Date
2017-03-28
Director of Research (if dissertation) or Advisor (if thesis)
Mineyev, Igor
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
M.S.
Degree Level
Thesis
Keyword(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.
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