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A Beauville surface is a complex algebraic surface that can be presented as a quotient of a product of two curves by a suitable action of a finite group. Bauer, Catanese and Grunewald have been able to intrinsically characterize the groups…

We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each…

Building on earlier results for regular maps and for orientably regular chiral maps, we classify the non-abelian finite simple groups arising as automorphism groups of maps in each of the 14 Graver-Watkins classes of edge-transitive maps.

In this paper, we prove a refinement of the Katsura theorem on finite group actions on abelian surfaces such that the quotient is birational to a $K3$ surface. As an application, we compute traces of Frobenius on the Neron--Severi groups of…

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of…

We show that if G is a group of automorphisms of a thick finite generalised quadrangle Q acting primitively on both the points and lines of Q, then G is almost simple. Moreover, if G is also flag-transitive then G is of Lie type.

We construct highly singular projective curves and surfaces defined by invariants of primitive complex reflection groups.

We describe complete simplicial toric varieties on which a unipotent group acts with a finite number of orbits. We also provide a complete list of such varieties in the case where the dimension is equal to 2.

We initiate the study of holomorphically convex groups: groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. If $G$ is a holomorphically convex group of…

We classify all spherical 2-designs that arise as orbits of finite group actions on real inner product spaces. Although it is well known that such designs can occur in representations without trivial components, we give a complete…

We give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of polarized abelian surfaces over finite fields.

We characterise the virtually abelian groups which are fundamental groups of compact K\"ahler manifolds and of smooth projective varieties. We show that a virtually abelian group is K\"ahler if and only if it is projective. In particular,…

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterium to deal with unipotent…

We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.

In this note, we consider all possible extensions G of a non-trivial perfect group H acting faithfully on a K3 surface X. The pair (X, G) is proved to be uniquely determined by G if the transcendental value of G is maximum. In particular,…

In this paper, we investigate automorphisms of compact K\"ahler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of…

We construct simple sharply 2-transitive groups. Our result answers an open question of Peter Neumann. In fact, we prove that every sharply 2-transitive group of characteristic 0 embeds into a simple sharply 2-transitive group.

We construct finitely generated simple torsion-free groups with strong homological control. Our main result is that every subset of $\mathbb{N} \cup \{\infty\}$, with some obvious exceptions, can be realized as the set of dimensions of…

We prove that the complex surfaces parametrizing cuboids and face cuboids, as well as their minimal resolution of singularities, have trivial fundamental group. We then compute the fundamental group of certain open smooth subvarieties of…

It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\frac{3}{2}$-generation, that every nontrivial element is…