Finding normal subgroups of the Cremona group

We study random walks on groups of isometries of non-proper delta-hyperbolic\nspaces under the assumption that at least one element in the group satisfies\nBestvina-Fujiwara's WPD condition. We show that in this case typical elements\nare WPD, and the Poisson boundary coincides with the Gromov boundary. Moreover,\nwe show that the random walk satisfies a form of asymptotic acylindricality,\nand we use this to show that the normal closure of random elements yields\nalmost surely infinitely many different normal subgroups. Moreover, the\nprobability that the normal closure is free tends to 1 if and only if the\nmaximal normal subgroup coincides with the center of the group. We apply such\ntechniques to the Cremona group, thus obtaining that the dynamical degree of\nrandom Cremona transformations grows exponentially fast, producing many\ndifferent normal subgroups, and identifying the Poisson boundary. We also give\na new identification of the Poisson boundary of Out(F_n). Our methods give\nbounds on the rates of convergence for these results.\n

Assuming the Borisov-Alexeev-Borisov conjecture, we prove that there is a constant $J=J(n)$ such that for any rationally connected variety $X$ of dimension $n$ and any finite subgroup $G\subset{\rm Bir}(X)$ there exists a normal abelian subgroup $A\subset G$ of index at most $J$. In particular, we obtain that the Cremona group ${\rm Cr}_3={\rm Bir}({\Bbb P}^3)$ enjoys the Jordan property.

For perfect fields k satisfying [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 that the Cremona group of rank n over (subfields of) the complex numbers is not simple for n≥3.

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane $ \\mathbb{P}_{\\mathbf{k}}^2 $ is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory and algebraic geometry to produce elements in the Cremona group that generate non-trivial normal subgroups.

Consider an algebraically closed field ${\textbf{k}}$, and let $\textsf{Cr}_2({\textbf{k}})$ be the Cremona group of all birational transformations of the projective plane over ${\textbf{k}}$. We characterize infinite order elements $g\in \textsf{Cr}_2({\textbf{k}})$ having a power $g^n$, $n\neq 0$, generating a proper normal subgroup of $\textsf{Cr}_2({\textbf{k}})$.

We prove that the group of birational transformations of a del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, the subgroup of Bir(ℙ 3 ) generated by all connected algebraic subgroups is a proper normal subgroup.

We study aspects of the group G_n of polynomial automorphisms of the affine space A^n, the so-called affine Cremona group. Shafarevich introduced on G_n the structure of an ind-variety, an infinite-dimensional analogon to a (classical) variety. The aim of this thesis is to study G_n within the framework of ind-varieties. The thesis consists of five articles. In the following we summarize them. 1. the Topologies on ind-Varieties and related Irreducibility Questions. In the literature there are two ways of endowing an affine ind-variety with a topology. One possibility is due to Shafarevich and the other due to Kambayashi. We specify a large class of affine ind-varieties where these two topologies differ. We give an example of an affine ind-variety that is reducible with respect to Shafarevich’s topology, but irreducible with respect to Kambayashi’s topology. Moreover, we give a counter-example of a supposed irreducibility criterion given by Shafarevich which is different from a counter-example given by Homma. We finish the article with an irreducibility criterion similar to the one given by Shafarevich. 2. Automorphisms of the Affine Cremona (joint with Hanspeter Kraft) We show that every automorphism of the group G_n is inner up to field automorphisms when restricted to the subgroup TG_n of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension n = 2 where all automorphisms are tame, i.e. TG_2 = G_2. The methods are different, based on arguments from algebraic group actions. 3. A Note on Automorphisms of the Affine Cremona Group Let G be an ind-group and let U be a unipotent ind-subgroup. We prove that an abstract automorphism f: G -> G maps U isomorphically onto a unipotent ind-subgroup of G, provided that f fixes a closed torus T in G that normalizes U and the action of T on U by conjugation fixes only the neutral element. As an application we generalize the main result of the article On Automorphisms of the Affine Cremona Group as follows: If an abstract automorphism of G_3 fixes the subgroup of tame automorphisms TG_3, then it also fixes a whole family of non-tame automorphisms (including the Nagata automorphism). 4. Automorphisms of the Plane Preserving a Curve (joint with Jeremy Blanc) We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the groups of positive dimension occuring is also given in the case where the curve is geometrically irreducible and the field is perfect. 5. Centralizer of a Unipotent Automorphism in the Affine Cremona Group Let g be a unipotent element of G_3. We describe the centralizer Cent(g) inside G_3. First, we treat the case when g is a modified translation. In the other case, we describe the subset Cent(g)_u of unipotent elements of Cent(g) and prove that it is a closed normal subgroup of Cent(g). Moreover, we show that Cent(g) is the semi-direct product of Cent(g)_u with a closed algebraic subgroup R of Cent(g). Finally, we prove that the subgroup of Cent(g) consisting of those elements that induce the identity on the algebraic quotient Spec O(A^3)^g form a characteristic subgroup of Cent(g).