Conjugacy, centralizers and dynamics in Thompson's group V

Thompson groups F, T and V were defined by Richard Thompson in connection to his work in logic in the process of building a group with unsolvable word problem and T and V provide the first known examples of finitely presented infinite simple groups. In this talk I will introduce this groups and focus on the last one. We explain how to solve the conjugacy problem and study the structure of centralizers, which are revealed by studying the dynamics of elements. We work with two equivalent element representations. The first one represents elements as pairs of trees and can be used to describe the dynamics. The second one sees elements as diagrams of strands which constitute a better combinatorial tool to understand conjugacy classes. In passing, we will review related results and other directions of this work. (Parts of this talk are joint work with others which will be appropriately cited).