Related papers: Linearization problem for finite subgroups of the …

We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some…

We study the completion of a group relative to a Zariski dense representation in a reductive algebraic group over a field $k$. The characteristic zero case was worked out previously by R. Hain; we extend his results to arbitrary…

We recall some properties, unfortunately not all, of the Cremona group. We first begin by presenting a nice proof of the amalgamated product structure of the well-known subgroup of the Cremona group made up of the polynomial automorphisms…

For perfect fields $k$ satisfying $[\bar k:k]>2$, we construct new normal subgroups of the plane Cremona group and provide an elementary proof of its non-simplicity, following the melody of the recent proof by Blanc, Lamy and Zimmermann…

We give a presentation of the plane Cremona group over an algebraically closed field with respect to the generators given by the Theorem of Noether and Castelnuovo. This presentation is particularly simple and can be used for explicit…

In this manuscript, we apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of characteristic zero. More precisely, we show that any linear algebraic…

We give the classification of the maximal infinite algebraic subgroups of the real Cremona group of the plane up to conjugacy and present a parametrisation space of each conjugacy class. Moreover, we show that the real plane Cremona group…

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane P^2 over k is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory, and…

We study the linearization problem of germs of holomorphic diffeomorphisms with resonant linear part. The formal linearization requires in general an infinite number of algebraic relations to be satisfied by the coefficients of the power…

We present a uniform methodology for computing with finitely generated matrix groups over any infinite field. As one application, we completely solve the problem of deciding finiteness in this class of groups. We also present an algorithm…

We prove that over any perfect field the plane Cremona group is generated by involutions.

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.

We study the properties of the fundamental group of an affine curve over an algebraically closed field of characteristic $p$, from the point of view of embedding problems. In characteristic zero, the fundamental group is free, but in…

We explore algebraic subgroups of of the Cremona group $\mathcal C_n$ over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of $\mathcal C_n$ that we call flattenable. It contains…

We give a way of estimating quickly the length of the hyperbolic isometry associated to a plane Cremona transformation that is presented as the product of two involutions. We also show, using the ideas of Cantat and Lamy, that, provided…

We present the abelianisation of the birational transformations of the real projective plane.

We characterize in terms of characteristic sequences the semigroups corresponding to branches at infinity of plane affine curves $\Gamma$ for which there exists a polynomial automorphism mapping $\Gamma$ onto the axis $x=0$.

The isomorphism problem means to decide if two given finite-dimensional simple algebras over the same centre are isomorphic and, if so, to construct an isomorphism between them. A solution to this problem has applications in computational…

The Cremona group is topologically simple when endowed with the Zariski or Euclidean topology, in any dimension $\ge 2$ and over any infinite field. Two elements are moreover always connected by an affine line, so the group is…

We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…