We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its Ellio...

Let G be the linear algebraic group SL3 over a field k of characteristic two. Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. We show that the full cohomology ring H(G,A) is finitely generated. This extends the finite generation property of the ring of invariants AG. We discuss where the problem stands for other geometrically reductive ...

We extend Haran’s Diamond Theorem to closed subgroups of a finitely generated free profinite group. This gives an affirmative answer to Problem 25.4.9 in [FrJ].

We prove that the abstract commensurator of a nontrivial free group, an infinite surface group, or more generally a group that splits appropriately over a cyclic subgroup is not finitely generated. This applies in particular to all torsion-free word-hyperbolic groups with infinite outer automorphism group and abelianization of rank at least 2. We also construct a finitely generated group which ...

The isomorphism problem for finitely generated Coxeter groups is the problem of deciding if two finite Coxeter matrices define isomorphic Coxeter groups. Coxeter [4] solved this problem for finite irreducible Coxeter groups. Recently there has been considerable interest and activity on the isomorphism problem for arbitrary finitely generated Coxeter groups. In this paper we describe a family of...

A new general formula for calculation of number of conjugacy classes of subgroups of given index in a finitely generated group is obtained.