Johannes Kunz

Title:	Homology of Thompson groups
Speaker:	Johannes Kunz
Time:	Thursday, 05.06.2025, 14:00
Place:	SR 3.69

Thompson’s groups $F\subset T \subset V$ were first introduced by R. J. Thompson in 1965 in unpublished handwritten notes. Since then, they have developed a live of their own, having an impact on a multitude of areas such as homotopy theory, logic, group theory, shape theory, dynamic data storage, etc.

An important aspect of these groups is their homology. While the homology of $F$ and $T$ is well known for almost 40 years, the homology of $V$ was first calculated in 2018 by Wahl and Szymik. They introduced a family of groups $\{V_{n,r}\}$ called Higman–Thompson groups, which generalize Thompson’s group $V$ . Moreover, they showed that $\{V_{n,r}\}$ satisfies homological stability. Using this, they showed that $V$ is integrally acyclic; that is, it has trivial homology, proving thereby a long standing hypothesis of Brown.

We will give a short overview of the ideas involved in calculating the homology of $F$ and $V$ . Therefore, we will in particular introduce cantor algebras and their automorphism groups, which are the Higman–Thompson groups mentioned above. Moreover, we will give a quick introduction to homological stability and how it is used to show the acyclicity of $V$ .