Title: Maximal subgroups in constant spinal groups

Abstract:
Groups of automorphisms of regular rooted trees are a rich source of
examples with interesting properties in group theory, and they have been
used to solve important problems. The first Grigorchuk group, defined by
Grigorchuk in 1980, is one of the first instances of an infinite
finitely generated periodic group, thus providing a negative solution to
the General Burnside Problem. It is also the first example of a group
with intermediate growth, hence solving the Milnor Problem. Many other
groups of automorphisms of rooted trees have since been defined and
studied. Important examples are the Gupta-Sidki group and the second
Grigorchuk group which belong to the  family of the so-called
Grigorchuk-Gupta-Sidki groups (GGS-groups, for short). Many
generalizations of the GGS-groups have also appeared in the recent
years, such as the large family of the so-called constant spinal groups.

A question that seams to appear repeatedly when working with groups of
automorphisms of regular rooted trees is weather a group is contained in
the class $\mathcal{MF}$ of groups with all maximal subgroups of finite
index.

The aim of this talk is to present a criterion that ensures that certain
constant spinal groups are contained in the class $\mathcal{MF}$, and
with the use of this criterion present some branch groups within
$\mathcal{MF}$ exhibiting novel properties, for example groups that
possess non-normal maximal subgroups. Furthermore, we also present some
tools that help to find new  examples of branch groups outside
$\mathcal{MF}$, with  maximal subgroups of infinite index.

This is a joint work with J. Moritz Petschick.