In this talk, we explore several zero-sum invariants for a finite group G: the small Davenport constant d(G), the ordered Davenport constant $Dₒ(G)$, and the Loewy length $L(G)$. We see that $Dₒ(G) = d(G) + 1 = d(A) + 2 $ for every non-abelian group of the form $G = A ⋊₋₁ C₂$, where $A$ is any finite abelian group. In addition, Dimitrov conjectured in 2004 that $Dₒ(G) = L(G)$ for every finite non-abelian p-group $G$. We provide explicit families of finite non-abelian p-groups for which Dimitrov's conjecture holds.

Let $V$ be a vector space over $C$ or a (right) module over the quaternions $H$, and let $G < GL(V)$ be a finite group generated by reflections. The hyperplane arrangement $A=A(G)$ given by the set of fixed spaces of the reflections in G has been an object of interest from different perspectives. For instance, for the symmetric group $G=S_n$ one gets the braid arrangement, whose complement space $M=M(A)$ is well-studied. In particular, its cohomology $H*(M) $ is well known (also as a representation), and the cohomology of the associated braid group is naturally given by the invariants $H*(M)^G$. In this talk I will present results on the latter object beyond real and complex reflection groups: Poincaré series of the invariants were computed in relation to generalized braid groups by Brieskorn for Coxeter groups, and for complex reflection groups in distinct papers by Lehrer, Callegaro-Marin, and finally Douglass-Pfeiffer-Röhrle. We study the same for finite quaternionic reflection groups, first classified by A.Cohen in 1980. This is joint work with Gerhard Röhrle and Johannes Schmitt.

Grigorchuk-Gupta-Sidki groups are key examples of groups acting on rooted trees, which are significant throughout the theory of infinite groups and beyond. There are very recent links between groups acting on rooted trees and algebraic geometry. Specifically, Gul and Uria-Albizuri showed in 2018 that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, admit Beauville structures. Based on recent work by Moritz Petschick, one can show that the periodic GGS-groups, that are defined by either an inverse-symmetric vector or a symmetric vector, have quotients that admit strongly real Beauville structures. Recall that a complex surface S is called real if there exists a biholomorphism between S and its complex conjugate surface such that the biholomorphism is an involution. The concept of strongly real is slightly more restrictive, and will be defined in the talk. This is joint work with Amir Dzambic.

TBA