The problem of determining all finite quotients of groups acting on rooted trees
Alejandra Garrido (Universidad Complutense de Madrid)
Monday 2nd March 15:00-16:00
Maths 311B
Abstract
A good way of studying infinite residually finite groups is through its finite quotients. The first problem one encounters is finding all finite quotients of the infinite residually finite group one is interested in.
A naive method is to take a "natural family" of finite quotients and hope that this accounts for all finite quotients. If this happens, the group is said to have the congruence subgroup property.
For example, if the group is a matrix group over the integers, the "natural family" is that obtained by taking entries of matrices modulo natural numbers (this is the context first studied, whence the name of the property).
A naive method is to take a "natural family" of finite quotients and hope that this accounts for all finite quotients. If this happens, the group is said to have the congruence subgroup property.
For example, if the group is a matrix group over the integers, the "natural family" is that obtained by taking entries of matrices modulo natural numbers (this is the context first studied, whence the name of the property).
In the case of groups acting on rooted trees, the "natural family" is given by the induced actions on finite rooted subtrees.
Self-similar groups are particularly nice groups of automorphisms of rooted trees, as the self-similarity makes calculations relatively easier than in the general setting.
Within this class, (weakly) branch groups are important because of their subgroup structure; they contain many direct products that encode the tree on which they act.
After reviewing some of the known results on the congruence subgroup property for various combinations of self-similar and (weakly) branch groups, I plan to talk on joint work with Zoran Sunic where we give a method of calculating certain obstructions to having the congruence subgroup property, for certain classes of self-similar branch groups.
Examples will be given. No previous knowledge of self-similar or branch groups is required.
Self-similar groups are particularly nice groups of automorphisms of rooted trees, as the self-similarity makes calculations relatively easier than in the general setting.
Within this class, (weakly) branch groups are important because of their subgroup structure; they contain many direct products that encode the tree on which they act.
After reviewing some of the known results on the congruence subgroup property for various combinations of self-similar and (weakly) branch groups, I plan to talk on joint work with Zoran Sunic where we give a method of calculating certain obstructions to having the congruence subgroup property, for certain classes of self-similar branch groups.
Examples will be given. No previous knowledge of self-similar or branch groups is required.
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