We prove the following. Theorem. Let π be the fundamental group of a finite graph of groups with finitely generated vertex groups Gv having asdimGv ≤ n for all vertices v. Then asdimπ ≤ n + 1. This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.

Conjugacy separability of any group of the class of one-relator groups given by the presentation a, b; a m , b n 1m, n > 1 is proven. The proof made used of theoretical combinatorial group methods, namely the structure of amalgamated free products and some properties of the subgroups and quotients of any group of the class of one-relator groups given above.

Let T be the quasi-nilpotent DT-operator. By use of Voiculescu’s amalgamated R-transform we compute the momets of (T −λ1)∗(T − λ1) where λ ∈ C, and the Brown-measure of T + √ Y , where Y is a circular element ∗-free from T for > 0. Moreover we give a new proof of Śniady’s formula for the moments τ(((T ∗)kT )) for k, n ∈ N.