Amir Džambić

AUTOMORPHISMS AND QUOTIENTS OF QUATERNIONIC FAKE QUADRICS

A Q-homology quadric is a normal projective algebraic surface with the same Betti numbers as the smooth quadric in P 3 . A smooth Q-homology quadric is either rational or of general type with vanishing geometric genus. Smooth minimal Q-homology quadrics of general type are called fake quadrics. Here we study quaternionic fake quadrics, that is, fake quadrics whose fundamental group is an irreducible lattice in PSL2.R/ PSL2.R/ derived from a division quaternion algebra over a real number field. We provide examples of quaternionic fake quadrics X with a nontrivial automorphism group G and compute the invariants of the quotient X= G and of its minimal desingularization Z. In this way we provide examples of singular Qhomology quadrics and minimal surfaces Z of general type with qD pgD 0 and K 2 D 4 or 2 which contain the maximal number of disjoint . 2/-curves. Conversely, we also show that if a smooth minimal surface of general type has the same invariant as Z and same number of . 2/-curves, then we can construct geometrically a surface of general type with c 2 D 8, c2D 4.

Macbeaths infinite series of Hurwitz groups

In the present paper we will construct an infinite series of so-called Hurwitz groups. One possible way to describe Hurwitz groups is to define them as finite homomorphic images of the Fuchsian triangle group with the signature (2, 3, 7). A reason why Hurwitz groups are interesting lies in the fact, that precisely these groups occur as the automorphism groups of compact Riemann surfaces of genus g > 1, which attain the upper bound 84(g − 1) for the order of the automorphism group. For a long time the only known Hurwitz group was the special linear group PSL2(\( \mathbb{F}_7 \)), with 168 elements, discovered by F. Klein in 1879, which is the automorphism group of the famous Kleinian quartic. In 1967 Macbeath found an infinite series of Hurwitz groups using group theoretic methods. In this paper we will give an alternative arithmetic construction of this series.

On the Cohomology of the Stover Surface

ABSTRACT We study a surface discovered by Stover which is the surface with minimal Euler number and maximal automorphism group among smooth arithmetic ball quotient surfaces. We study the natural map and discuss the problem related to the so-called Lagrangian surfaces. We obtain that this surface S has maximal Picard number and has no higher genus fibrations. We compute that its Albanese variety A is isomorphic to , for α = e2iπ/3.

Invariants of some compactified Picard modular surfaces and applications

The paper investigates invariants of compactified Picard modular surfaces by principal congruence subgroups of Picard modular groups. The applications to the surface classification and modular forms are discussed.