Related papers: Cremona groups of real surfaces

We bound the indices of normal abelian subgroups in finite groups contained in the Cremona group of rank 2 over a field of odd characteristic.

We look at algebraic embeddings of the Cremona group in $n$ variables $Cr_n(C)$ to the group of birational transformations $Bir(M)$ of an algebraic variety $M$. First we study geometrical properties of an example of an embedding of…

Let $X$ be a compact Riemann surface of genus $g$ and let $x \in X$. We derive the classical presentation of $\pi_1(X,x)$ (i.e the one given by $2g$ generators $a_1,b_1, \dots, a_g,b_g$ and the relation $\prod_{i=1}^g[a_i,b_i] = 1$) from…

We construct first examples of discrete geometrically finite subgroups of PU(2,1) which contain parabolic elements, and are isomorphic to surface groups.

The Cremona group is connected in any dimension and, endowed with its topology, it is simple in dimension 2. ----- Le groupe de Cremona est connexe en toute dimension et, muni de sa topologie, il est simple en dimension 2.

Matrix generators for the general and special linear groups, the symplectic groups and the general and special unitary groups over finite fields. For the most part the generators have been obtained by translating Steinberg's generators for…

We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, L. Bayle and A. Beauville), which…

This expository article builds on lecture notes from a minicourse entitled "Cremona groups and CAT(0) cube complexes" and given by the author as part of the 2023 Riverside Workshop on Geometric Group Theory. It presents recent constructions…

We construct examples of algebraic surfaces with interesting fundamental groups.

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is…

Let Cr(k) be the Cremona group of rank 2 over a field k, i.e. the group of all k-automorphisms of k(X,Y). We determine the l.c.m. of the orders of the finite subgroups of Cr(k) of order prime to the characteristic of k.

In order to enumerate the fake projective planes, as announced in~\cite{CS}, we found explicit generators and a presentation for each maximal arithmetic subgroup $\bar\Gamma$ of~$PU(2,1)$ for which the (appropriately normalized) covolume…

Let $\mathbf{K}$ be an algebraically closed field. The Cremona group $\operatorname{Cr}_2(\mathbf{K})$ is the group of birational transformations of the projective plane $\mathbb{P}^2_{\mathbf{K}}$. We carry out an overall study of…

In this paper we shall be looking at several results relating Schur rings to sufficient conditions for a graph to be a graphical regular representation (GRR) of a finite group, and then applying these specifically in the case of certain…

We give a method for constructing many examples of automorphisms with positive entropy on rational complex surfaces. The general idea is to begin with a quadratic Cremona transformation that fixes a reduced cubic curve and then use the…

We consider iterated preimages of curves by random products of birational transformations of the plane. Following a recent work of Diller and Roeder, we study the action of the Cremona group on the inverse limit of the spaces of currents in…

We construct embeddings of surface groups into the group of germs of analytic diffeomorphisms in one variable.

In this short note, we extend the results of [Alexeev-Orlov, 2012] about Picard groups of Burniat surfaces with $K^2=6$ to the cases of $2\le K^2\le 5$. We also compute the semigroup of effective divisors on Burniat surfaces with $K^2=6$.…

The existence of an infinite simple boundedly generated 2-generated group and the existence of a boundedly simple 2-generated group containing a free non-cyclic subgroup are proved.

We present almost complete list of normal forms of classical $r$-matrices on the Poincar\'{e} group.