Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations

TL;DR: In this article, the algebraic structure of finitely generated groups acting freely and regularly on Zn-trees and necessary and sufficient conditions for such actions are given, along with a robust algorithmic theory.

TL;DR: In this article, higher order regionally proximal relations suitable for an arbitrary acting group were introduced and shown to be equivalence relations whenever the action is minimal, and these relations lifted through extensions between minimal systems.

TL;DR: In this paper, the authors studied the action of the absolute Galois group Gal(Q/Q) on dessins d'enfants and Beauville surfaces and proved that the action is faithful on the set of quasiplatonic (or triangle) curves of any given hyperbolic type.

TL;DR: In this article, the authors proposed a conceptual explanation based on Kazhdan's property (T) from representation theory of Lie groups and improved some of the previous estimates on the running time of the product replacement algorithm.