The aim of the present note is to show that free Burnside groups of sufficiently large odd exponent are non–amenable in a certain strong sense, more precisely, their left regular representations are isolated from the trivial representation uniformly on finite generating sets. This result is applied to the solution of a strong version of the von Neumann – Day problem concerning amenability of gr...

We discuss the equivariant Burnside group and related new invariants in birational geometry, with a special emphasis on applications low dimensions.

We consider the problem of sampling “unlabelled structures”, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible...

Let G be a finite group and let k be a field. Our purpose is to investigate the simple modules for the double Burnside ring kB(G,G). It turns out that they are evaluations at G of simple biset functors. For a fixed finite group H, we introduce a suitable bilinear form on kB(G,H) and we prove that the quotient of kB(−, H) by the radical of the bilinear form is a semi-simple functor. This allows ...

The notion of amenability was introduced in 1929 by J. von Neumann [68] in order to explain the Banach-Tarski paradox. A countable discrete group Γ is amenable if there exists a left-invariant mean φ : `∞(Γ) → C. The class of amenable groups is stable under subgroups, direct limits, quotients and the free group F2 on two generators is not amenable. Knowing whether or not the class of amenable g...

We study 2-representations, i.e. actions of 2-groups on 2-vector spaces. Our main focus is character theory for 2-representations. To this end we employ the technique of extended Burnside rings. Our main theorem is that the Ganter-Kapranov 2-character is a particular mark homomorphism of the Burnside ring. As an application we give a new proof of Osorno formula for the Ganter-Kapranov 2-charact...

This paper represents a finite group with 32 elements as a group of transformations of a compact surface of genus 5. In particular, we start with a designated pair of regions of this surface, and each region is labeled with the group element, which transforms the designated region into it. This gives a portrait of that finite group. These surfaces and the regions corresponding to the group elem...

1 Let G be a finite group. The Burnside ring B(G) of the group G is one of the fundamental representation rings of G, namely the ring of permutation representations. It is in many ways the universal object to consider when looking at the category of G-sets. It can be viewed as an analogue of the ring Z of integers for this category. It can be studied from different points of view. First B(G) is...

We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigo...

By studying the so-called traveling salesman groups, we obtain a new metric criterion for non-amenability. As an application, we give a new and very short proof of non-amenability of free Burnside groups with sufficiently big odd exponent. Amenable groups have been introduced by John von Neumann in 1929 in connection with Banach-Tarski Paradox, although earlier, Banach himself had understood th...