Title:

The width of verbal subgroups in profinite groups

The main result of this thesis is an original proof that every word has finite width in a compact $p$adic analytic group. The proof we give here is an alternative to Andrei JaikinZapirain's recent proof of the same result, and utilises entirely grouptheoretical ideas. We accomplish this by reducing the problem to a proof that every word has finite width in a profinite group which is virtually a polycyclic pro$p$ group. To obtain this latter result we first establish that such a group can be embedded as an open subgroup of a group of the form $N_1M_1$, where $N_1$ is a finitely generated closed normal nilpotent subgroup, and $M_1$ is a finitely generated closed nilpotentbyfinite subgroup; we then adapt a method of V. A. Romankov. As a corollary we note that our approach also proves that every word has finite width in a polycyclicbyfinite group (which is not profinite). As a supplementary result we show that for finitely generated closed subgroups $H$ and $K$ of a profinite group the commutator subgroup $[H,K]$ is closed, and give examples to show that various hypotheses are necessary. This implies that the outercommutator words have finite width in profinite groups of finite rank. We go on to establish some bounds for this width. In addition, we show that every word has finite width in a product of a nilpotent group of finite rank and a virtually nilpotent group of finite rank. We consider the possible application of this to soluble minimax groups.
