The width m(w, G) of a word w in a group G is the maximal distance to 1 of any element of the Cayley graph of the verbal subgroup w(G) with respect to the natural generating set of all w-values G_{w}. An open problem is to characterise the words w where for all finite groups G the growth of m(w, G) is bounded in terms of the rank of G. Our goal is to understand the solution of this problem for finite nilpotent groups, as given in the exposition [1].
Let G be a finitely generated residually finite group and denote a_{n}(G) the number of subgroups of G of index n. The subgroup growth is the study of the asymptotic behaviour of the sequence (a_{n}(G))_{n ∈ ℕ} and it turned out that this holds a wealth of information about G. Analogously one can define the sequence for normal subgroups that is denoted by a^{▹}_{n} ( G ). The aim of the seminar is to present 6 self-contained talks around this topic. We will see some results that were not covered during the GRK lectures and some explicit computations.
Mixed Topics
Word Growth in Groups
Bass-Serre Theory and Profinite Analogues
p-Adic analytic pro-p groups
Invariant random subgroups
Probabilistic methods in group theory
Buildings