Volume 19, Issue 1

In the modular representation theory of finite groups, we show that the standard derivation of the Green correspondence lifts to a derivation of a Green correspondence for twisted group algebras (Theorem 1.3). Then, from these results we derive a lift of the Puig correspondences for twisted group algebras (Theorem 1.6). Clearly twisted group algebras arise naturally in finite group modular representation theory. We conclude with some suggestions for applications in this mathematical area.

The Stallings construction for f.g. subgroups of free groups is generalized by introducing the concept of Stallings section, which allows efficient computation of the core of a Schreier graph based on edge folding. It is proved that the groups that admit Stallings sections are precisely the f.g. virtually free groups, this is proved through a constructive approach based on Bass–Serre theory. Complexity issues and applications are also discussed.

Two trees in the boundary of outer space are said to be primitive-equivalent whenever their translation length functions are equal in restriction to the set of primitive elements of F N . We give an explicit description of this equivalence relation, showing in particular that it is nontrivial. This question is motivated by our description of the horoboundary of outer space for the Lipschitz metric in [`The horoboundary of outer space, and growth under random automorphisms', preprint (2014)]. In proving this, we extend a theorem due to White about the Lipschitz metric on outer space to trees in the boundary, showing that the infimal Lipschitz constant of an F N -equivariant map between the metric completion of any two minimal, very small F N -trees is equal to the supremal ratio between the translation lengths of the elements of F N in these trees. We also provide approximation results for trees in the boundary of outer space.

Let Γ be a finitely generated nilpotent group and let G be a complex reductive algebraic group. The representation variety Hom(Γ, G ) and the character variety Hom(Γ, G )// G each carry a natural topology, and we describe the topology of their connected components in terms of representations factoring through quotients of Γ by elements of its lower central series.

We show that a finitely generated subgroup of a free group, chosen uniformly at random, is strictly Whitehead minimal with overwhelming probability. Whitehead minimality is one of the key elements of the solution of the orbit problem in free groups. The proofs strongly rely on combinatorial tools, notably those of analytic combinatorics. The result we prove actually depends implicitly on the choice of a distribution on finitely generated subgroups, and we establish it for the two distributions which appear in the literature on random subgroups.

The Andrews–Curtis conjecture asserts that, for a free group F n of rank n and a free basis ( x 1 ,..., x n ), any normally generating tuple ( y 1 ,..., y n ) is Andrews–Curtis equivalent to ( x 1 ,..., x n ). This equivalence corresponds to the actions of Aut F n and of F n n on normally generating n -tuples. The equivalence corresponding to the action of Aut F n on generating n -tuples is called Nielsen equivalence. The conjecture for arbitrary finitely generated groups has its own importance to analyse potential counter-examples to the original conjecture. We study the Andrews–Curtis and Nielsen equivalence in the class of finitely generated groups for which every maximal subgroup is normal, including nilpotent groups and Grigorchuk groups.

In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? An affirmative answer is given for all compact groups with the exception of some metric profinite groups which are almost perfect and strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup.