# Contributions to automorphisms of affine spaces

Stampfli, Immanuel. Contributions to automorphisms of affine spaces. 2013, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_10504

## Abstract

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. On 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. On Automorphisms of the Affine Cremona Group (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 Déserti 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 Jérémy 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).
Advisors: Kraft, Hanspeter Dubouloz, A. 05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Algebra (Kraft) Stampfli, Immanuel and Kraft, Hanspeter Thesis Doctoral Thesis 10504 Complete Link to catalogue 79 S. English doi: 10.5451/unibas-006173431urn: urn:nbn:ch:bel-bau-diss105044 22 Jan 2018 15:51 22 Oct 2013 12:49

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