These are the lecture notes of a 2-hour mini-course on Lie groups over local fields presented at the Workshop on Totally Disconnected Groups, Graphs and Geometry at the Heinrich-Fabri-Institut Blaubeuren in May 2007. The goal of the notes is to pro
vide an introduction to p-adic Lie groups and Lie groups over fields of formal Laurent series, with an emphasis on relations to the structure theory of totally disconnected, locally compact groups. In particular, they contain a discussion of the scale, tidy subgroups and contraction groups for automorphisms of Lie groups over local fields. Special attention is paid to the case of Lie groups over local fields of positive characteristic.
In a 2004 article, Udo Baumgartner and George Willis used ideas from the structure theory of totally disconnected, locally compact groups to achieve a better understanding of the contraction group U_f associated with an automorphism f of such a group
G, assuming that G is metrizable. (Recall that U_f consists of all group elements x such that f^n(x) tends to the identity element as n tends to infinity). Recently, Wojciech Jaworski showed that the main technical tool of the latter article remains valid in the non-metrizable case. He asserted without proof that, therefore, all results from that article remain valid. However, metrizability enters the arguments at a second point. In this note, we resolve this difficulty, by providing an affirmative answer to a question posed by Willis in 2004.
We give an exposition of the theory of invariant manifolds around a fixed point, in the case of time-discrete, analytic dynamical systems over a complete ultrametric field K. Typically, we consider an analytic manifold M modelled on an ultrametric
Banach space over K, an analytic self-map f of M, and a fixed point p of f. Under suitable conditions on the tangent map of f at p, we construct a centre-stable manifold, a centre manifold, respectively, an r-stable manifold around p, for a given positive real number r not exceeding 1. The invariant manifolds are useful in the theory of Lie groups over local fields, where they allow results to be extended to the case of positive characteristic which previously were only available in characteristic zero (i.e., for p-adic Lie groups).
Many infinite-dimensional Lie groups of interest can be expressed as a union of an ascending sequence of (finite- or infinite-dimensional) Lie groups. In this survey article, we compile general results concerning such ascending unions, describe the m
ain classes of examples, and explain what the general theory tells us about these. In particular, we discuss: (1) Direct limit properties of ascending unions of Lie groups in the relevant categories; (2) Regularity in Milnors sense; (3) Homotopy groups of direct limit groups and of Lie groups containing a dense union of Lie groups; (4) Subgroups of direct limit groups; (5) Constructions of Lie group structures on ascending unions of Lie groups.
We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb: (1) There exists a subgroup of some infinite-dimensional Lie group G which does not admit an initial Lie subgroup structure; (
2) The pathology cannot occur if G is a direct limit of an ascending sequence of finite-dimensional Lie groups; (3) Every such direct limit group is a ``topological group with Lie algebra in the sense of Hofmann and Morris. Moreover, we prove a version of Borels Theorem announced in the survey, ensuring the existence of compactly supported smooth diffeomorphisms with given Taylor series around a fixed point p (provided the tangent map at p has positive determinant).
In an earlier paper, we studied solutions g to convolution equations of the form a_d*g^{*d}+a_{d-1}*g^{*(d-1)}+...+a_1*g+a_0=0, where a_0, ..., a_d are given arithmetic functions associated with Dirichlet series which converge on some right half plan
e, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form sum_{xin X} f(x) e^{-sx} (s in C^k), where X is an additive subsemigroup of [0,infty)^k. If X is discrete and a certain solvability criterion is satisfied, we determine solutions by an elementary recursive approach, adapting an idea of Feckan. The solution of the general case leads us to a more comprehensive question: Let X be an additive subsemigroup of a pointed, closed convex cone C in R^k. Can we find a complex Radon measure on X whose Laplace transform satisfies a given polynomial equation whose coefficients are Laplace transforms of such measures?
This note is devoted to the theory of projective limits of finite-dimensional Lie groups, as developed in the recent monograph ``The Lie Theory of Connected Pro-Lie Groups by K.H. Hofmann and S.A. Morris. We replace the original, highly non-trivial p
roof of the One-Parameter Subgroup Lifting Lemma given in the monograph by a shorter and more elementary argument. Furthermore, we shorten (and correct) the proof of the so-called Pro-Lie Group Theorem, which asserts that pro-Lie groups and projective limits of Lie groups coincide.
Let G be a Lie group over a local field of positive characteristic which admits a contractive automorphism f (i.e., the forward iterates f^n(x) of each group element x converge to the neutral element 1). We show that then G is a torsion group of fini
te exponent and nilpotent. We also obtain results concerning the interplay between contractive automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon. Some of the results even extend to Lie groups over arbitrary complete ultrametric fields.
