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

Author(s): | Zhu, Kejia |

Advisor(s): | Mineyev, Igor |

Department / Program: | Mathematics |

Discipline: | Mathematics |

Degree Granting Institution: | University of Illinois at Urbana-Champaign |

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: | 2017-03-28 |

Type: | Thesis |

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

Rights Information: | Copyright 2017 Kejia Zhu |

Date Available in IDEALS: | 2017-08-10 |

Date Deposited: | 2017-05 |