Compactness of the space of marked groups and examples of L2-Betti numbers of simple groups

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.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Zhu, Kejia

Contributors dc:contributor

• Mineyev, Igor

• Geometric group theory
• Topology

Statement dc:rights

• Copyright 2017 Kejia Zhu

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/97234

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/97234