**SEMINARIO TEORÍA DE GRUPOS**

Portraits and patterns in contracting groups

**Ponente:** Jone Uria-Albizuri (Universidad del País Vasco)

**Fecha:** Jueves, 7 de abril de 2022 - 11:30

**Lugar:** Aula Naranja, ICMAT

**Resumen:**

Any automorphism of a $d$-regular rooted tree $T$ can be described by decorating the tree vertices with permutations from the symmetric group $S_d$. If one cuts this decoration at a certain level of the tree, we obtain a finite pattern.

On the other hand, if the action of a group $G$ on $T$ is what is known as contracting, each group element can be described by a finite decoration made up of a pattern on the top levels and group elements from a finite set on the leaves.

Given a contracting group, a natural question to ask is which are the patterns and portraits seen on the group. We point out some relations between these two notions, look at their growth for regular branch groups and analyse the situation in some particular examples.