Expansive automorphisms of totally disconnected, locally compact groups
| Autor: | C. R. E. Raja, Helge Glöckner |
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| Rok vydání: | 2016 |
| Předmět: |
Normal subgroup
Algebra and Number Theory 010102 general mathematics Neighbourhood (graph theory) Dynamical Systems (math.DS) Group Theory (math.GR) Locally compact group Automorphism 01 natural sciences Combinatorics Mathematics::Group Theory Nilpotent Totally disconnected space 0103 physical sciences FOS: Mathematics 010307 mathematical physics Locally compact space Mathematics - Dynamical Systems 0101 mathematics Quotient group Mathematics - Group Theory 22D05 (primary) 22D45 22E20 37A25 37P20 (secondary) Mathematics |
| Zdroj: | Journal of Group Theory. 20:589-619 |
| ISSN: | 1435-4446 1433-5883 |
| DOI: | 10.1515/jgth-2016-0051 |
| Popis: | We study automorphisms $\alpha$ of a totally disconnected, locally compact group $G$ which are expansive in the sense that, for some identity neighbourhood $U$, the sets $\alpha^n(U)$ (for integers $n$) intersect in the trivial group. Notably, we prove that the automorphism induced by $\alpha$ on $G/N$ for an $\alpha$-stable closed normal subgroup $N$ of $G$ is always expansive. Further results involve the associated contraction groups $U_\alpha$ consisting of all $x$ in $G$ such that $\alpha^n(x) \to e$ as $n$ tends to infinity. If $\alpha$ is expansive, then $W := U_\alpha U_{\alpha^{-1}}$ is an open identity neighbourhood in $G$. We give examples where $W$ fails to be a subgroup. However, $W$ is a nilpotent open subgroup whenever $G$ is a closed subgroup of a general linear group over the $p$-adic numbers. Further results are devoted to the divisible and torsion parts of $U_\alpha$, and to the so-called "nub" $U_0$ of an expansive automorphism $\alpha$ (the intersection of the closures of $U_\alpha$ and $U_{\alpha^{-1}}$). Comment: LaTeX, 32 pages; v3: minor improvements |
| Databáze: | OpenAIRE |
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