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

Geometric group theory;Topology

Zhu, Kejia ; Mineyev ; Igor

关键词: Geometric group theory;    Topology;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/97234/ZHU-THESIS-2017.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.