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.

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.

The representation growth of a group $G$ measures the asymptotic distribution of its irreducible representations. Whenever the growth is polynomial, a suitable vehicle for studying it is a Dirichlet generating series called the representation zeta function of $G$. One of the key invariants in this context is the abscissa of convergence of the representation zeta function. The spectrum of all abscissae arising across a given class of groups is of considerable interest and has been studied in some cases.

In the realm of $p$-adic analytic groups (with perfect Lie algebra), the abscissae of convergence are explicitly known only for groups of small dimensions. But there are interesting asymptotic results for 'simple' $p$-adic analytic groups of increasing dimensions. In this talk, I will give an overview of the main tools and ingredients in this area and I will report on recent work joint with Moritz Petschick to enlarge the class of groups for which we have explicit results.