Mathematics

In a pervious paper Weidmann shows that there a bound on the number of orbits of edges in a tree on which a finitely generated group acts (k,C) -acylindrically. In this paper we extend this result to actions which are k -acylindrical except on a family of groups with "finite height". We also give an example which gives a negative result to a conjecture of Weidmann from the same paper and produce a sharp bound for groups acting k --acylindrically.

In this paper, we show that every irreducible 2 -dimensional Artin group A Γ of rank at least 3 is acylindrically hyperbolic. We do this by applying a criterion of Martin to the action of A Γ on its modified Deligne complex. Along the way, we prove results of independent interests on the geometry of links of this complex.

In our article in MCU'2013 we state the the Domino problem is undecidable for all Baumslag-Solitar groups BS(m,n) , and claim that the proof is a direct adaptation of the construction of a weakly aperiodic subshift of finite type for BS(m,n) given in the paper. In this addendum, we clarify this point and give a detailed proof of the undecidability result. We assume the reader is already familiar with the article in MCU'2013.

Let Λ be an ordered abelian group, Aut + (Λ) the group of order-preserving automorphisms of Λ , G a group and α:G→ Aut + (Λ) a homomorphism. An α -affine action of G on a Λ -tree X is one that satisfies d(gx,gy)= α g d(x,y) ( x,y∈X , g∈G ). We consider classes of groups that admit a free, rigid, affine action in the case where X=Λ . Such groups form a much larger class than in the isometric case. We show in particular that unitriangular groups UT(n,R) and groups T ∗ (n,R) of upper triangular matrices over R with positive diagonal entries admit free affine actions. Our proofs involve left symmetric structures on the respective Lie algebras and the associated affine structures on the groups in question. We also show that given ordered abelian groups Λ 0 and Λ 1 and an orientation-preserving affine action of G on Λ 0 , we obtain another such action of the wreath product G≀ Λ 1 on a suitable Λ ′ . It follows that all free soluble groups, residually free groups and locally residually torsion-free nilpotent groups admit essentially free affine actions on some Λ ′ .

The 15-Puzzle is a well studied permutation puzzle. This paper explores the group structure of a three-dimensional variant of the 15-Puzzle known as the Varikon Box, with the goal of providing a heuristic that would help a human solve it while minimizing the number of moves. First, we show by a parity argument which configurations of the puzzle are reachable. We define a generating set based on the three dimensions of movement, which generates a group that acts on the puzzle configurations, and we explore the structure of this group. Finally, we show a heuristic for solving the puzzle by writing an element of the symmetry group as a word in terms of a generating set, and we compute the shortest possible word for each puzzle configuration.

Let G be the alternating group Alt(n) on n letters. We prove that for any ε>0 there exists N=N(ε)∈N such that whenever n≥N and A , B , C , D are normal subsets of G each of size at least |G | 1/2+ε , then ABCD=G .

Let M be a cancellative monoid. It is known~\cite{Ta54} that if M is left amenable then the monoid ring K[M] satisfies Ore condition, that is, there exist nontrivial common right multiples for the elements of this ring. In~\cite{Don10} Donnelly shows that a partial converse to this statement is true. Namely, if the monoid Z + [M] of all elements of Z[M] with positive coefficients has nonzero common right multiples, then M is left amenable. He asks whether the converse is true for this particular statement. We show that the converse is false even for the case of groups. If M is a free metabelian group, then M is amenable but the Ore condition fails for Z + [M] . Besides, we study the case of the monoid M of positive elements of R.\,Thompson's group F . The amenability problem for it is a famous open question. It is equivalent to left amenability of the monoid M . We show that for this case the monoid Z + [M] does not satisfy Ore condition. That is, even if F is amenable, this cannot be shown using the above sufficient condition.

In this paper, we show that each finite group G containing at most p 2 Sylow p -subgroups for each odd prime number p , is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.

In this article, we study connections between components of the Cayley graph Cay(G,A) , where A is an arbitrary subset of a group G , and cosets of the subgroup of G generated by A . In particular, we show how to construct generating sets of G if Cay(G,A) has finitely many components. Furthermore, we provide an algorithm for finding minimal generating sets of finite groups using their Cayley graphs.

We show that the groups of finite energy loops and paths (that is, those of Sobolev class H 1 ) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation π of such a group (which we call skew-amenable) has a conjugation-invariant state on B( H π ) .