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  • Autor: Helge Glöckner x
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Abstract

Let G be a totally disconnected, locally compact group admitting a contractive automorphism α. We prove a Jordan-Hölder theorem for series of α-stable closed subgroups of G, classify all possible composition factors and deduce consequences for the structure of G.

Abstract

We analyze the structure of locally compact groups which can be built up from p-adic Lie groups, for p in a given set of primes. In particular, we calculate the scale function and determine tidy subgroups for such groups, and use them to recover the primes needed to build up the group.

Abstract

We show that every finite-dimensional p-adic Lie group of class Ck (where k ∈ ℕ ∪ {∞}) admits a Ck-compatible analytic Lie group structure. We also construct an exponential map for every k + 1 times strictly differentiable (SC k+1) ultrametric p-adic Banach-Lie group, which is an SC 1-diffeomorphism and admits Taylor expansions of all finite orders ≤ k.

Abstract

We study topological automorphisms α of a totally disconnected, locally compact group G which are expansive in the sense that

nαn(U)={1}

for some identity neighbourhood UG. Notably, we prove that the automorphism induced by an expansive automorphism α on a quotient group G/N modulo an α-stable closed normal subgroup N is always expansive. Further results involve the contraction groups

Uα:={gG:αn(g)1 as n}.

If α is expansive, then UαUα-1 is an open identity neighbourhood in G. We give examples where UαUα-1 fails to be a subgroup. However, UαUα-1 is an α-stable, nilpotent open subgroup of G if G is a closed subgroup of GLn(p). Further results are devoted to the divisible and torsion parts of Uα, and to the so-called “nub” nub(α)=Uα¯Uα-1¯ of an expansive automorphism.