Abstracts Category : Other

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

by Kejia Zhu

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 0 there exists an infinite simple group Q generated by n elements such that (2),(Q) n 1 . 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 doesnt affect the final result.Advisors/Committee Members: Mineyev, Igor (advisor).