On the congruence subgroup problem for branch groups

We answer a question of Bartholdi, Siegenthaler and Zalesskii, showing that the congruence subgroup problem for branch groups is independent of the branch action on a tree. We prove that the congruence topology of a branch group is determined by the group; specifically, by its structure graph, an object first introduced by Wilson. We also give a more natural definition of this graph.Comment: 9 pages, no figures; minor changes in accordance with referee report, exposition improve

Embedded antennas for signal-transmissive-walls in radio-connected low-energy urban buildings

The work presented in this thesis intends to develop and analyze two solutions to improve the 5G signal indoor coverage in the Finnish buildings. Two frequency ranges of the 5G spectrum are considered relevant, sub-6,GHz and ac{mmWave}. A through analysis is carried out, from the description to the simulation of the proposed solutions.Outgoin

Branch groups with infinite rigid kernel

A theoretical framework is established for explicitly calculating rigid kernels of self-similar regular branch groups. This is applied to a new infinite family of branch groups in order to provide the first examples of self-similar, branch groups with infinite rigid kernel. The groups are analogs of the Hanoi Towers group on 3 pegs, based on the standard actions of finite dihedral groups on regular polygons with odd numbers of vertices, and the rigid kernel is an infinite Cartesian power of the cyclic group of order 2, except for the original Hanoi group. The proofs rely on a symbolic-dynamical approach, related to finitely constrained groups.Comment: Comments welcome

Free factors and profinite completions

Can one detect free products of groups via their profinite completions? We answer positively among virtually free groups. More precisely, we prove that a subgroup of a finitely generated virtually free group $G$ is a free factor if and only if its closure in the profinite completion of $G$ is a profinite free factor. This generalises results by Parzanchevski and Puder (later also proved by Wilton) for free groups. Our methods are entirely different to theirs, combining homological properties of profinite groups and the decomposition theory of Dicks and Dunwoody.Comment: 21 pages, accepted for publication in IMR

Discrete locally finite full groups of Cantor set homeomorphisms

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a.~ topological full groups). We determine that any piecewise full group that is locally compact in the compact-open topology on the group of self-homeomorphisms of the Cantor set must be uniformly discrete, in a precise sense that we introduce here. Uniformly discrete groups of self-homeomorphisms of the Cantor set are in particular countable, locally finite, residually finite and discrete in the compact-open topology. The resulting piecewise full groups form a subclass of the ample groups introduced by Krieger. We determine the structure of these groups by means of their Bratteli diagrams and associated dimension ranges ($K_0$ groups). We show through an example that not all uniformly discrete piecewise full groups are subgroups of the ``obvious'' ones, namely, piecewise full groups of finite groups.Comment: 18 pages, 6 figures, accepted version for publicatio