The class of automaton groups is a rich source of the simplest examples of infinite Burnside groups. However, there are some classes of automata that do not contain such examples. For instance, all infinite Burnside automaton groups in the literature are generated by non reversible Mealy automata and it was recently shown that 2-state invertible-reversible Mealy automata cannot generate infinit...

The Brauer group of a commutative ring is an important invariant of a commutative ring, a common journeyman to the group of units and the Picard group. Burnside rings of finite groups play an important rôle in representation theory, and their groups of units and Picard groups have been studied extensively. In this short note, we completely determine the Brauer groups of Burnside rings: they van...

Let G be a finite group. It is well known that a Mackey functor {H 7→ M(H)} is a module over the Burnside ring functor {H 7→ Ω(H)}, where H ranges over the set of all subgroups of G. For a fixed homomorphism w : G → {−1, 1}, the Wall group functor {H 7→ Ln(Z[H], w|H)} is not a Mackey functor if w is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside rin...

In this work we investigate the hardness of a computational problem introduced in the recent work of Baumslag et al. in [3, 4]. In particular, we study the Bn-LHN problem, which is a generalized version of the learning with errors (LWE) problem, instantiated with a particular family of non-abelian groups (free Burnside groups of exponent 3). In our main result, we demonstrate a random self-redu...

Let G be a finite group, and R be a commutative ring. This note proposes a generalization to any Green functor for G over R of the construction of the Hochschild cohomology ring HH∗(G,R) from the ordinary cohomology functor H∗(−, R). Another special case is the construction of the crossed Burnside ring of G from the ordinary Burnside functor. The general abstract setting is the following : let ...

In these notes we detail the geometrical approach of small cancellation theory used by T. Delzant and M. Gromov to provide a new proof of the infiniteness of free Burnside groups and periodic quotients of torsion-free hyperbolic groups.