A periodicity theorem for acylindrically hyperbolic groups

We generalize a well-known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization has been used to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.

On Knörrer Periodicity for Quadric Hypersurfaces in Skew Projective Spaces

We study the structure of the stable category $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ of graded maximal Cohen–Macaulay module over $S/(f)$ where $S$ is a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree 1, and $f=x_{1}^{2}+\cdots +x_{n}^{2}$. If $S$ is commutative, then the structure of $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is well known by Knörrer’s periodicity theorem. In this paper, we prove that if $n\leqslant 5$, then the structure of $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is determined by the number of irreducible components of the point scheme of $S$ which are isomorphic to $\mathbb{P}^{1}$.

FINE AND WILF'S THEOREM FOR k-ABELIAN PERIODS

Two words u and v are k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. This leads to a hierarchy of equivalence relations on words which lie properly in between the equality and abelian equality. The goal of this paper is to analyze Fine and Wilf's periodicity theorem with respect to these equivalence relations. Fine and Wilf's theorem tells exactly how long a word with two periods p and q can be without having the greatest common divisor of p and q as a period. Recently, the same question has been studied for abelian periods. In this paper we show that for k-abelian periods the situation is similar to the abelian case: In general, there is no bound for the lengths of such words, but the values of the parameters p, q and k for which the length is bounded can be characterized. In the latter case we provide nontrivial upper and lower bounds for the maximal lengths of such words. In some cases (e.g., for k = 2) we found the maximal length precisely.