Infinite subgroups of the Morava stabilizer groups
type: Item
instance of: scholarly article
title: Infinite subgroups of the Morava stabilizer groups
zbMATH Open document ID: 0937.16026
DOI: 10.1016/S0040-9383(97)00100-6
author: Peter Symonds
author: V. Gorbounov
author: Mark Mahowald
published in: Topology
publication date: 2000-06-06T00:00:00Z
review text: Let \(D\) be a division algebra over the \(p\)-adic field \(\mathbb{Q}_p\), of index \(n\) and Hasse invariant \(\tfrac 1n\). Write \(W\) for the totally unramified extension of \(\mathbb{Q}_p\) of degree \(n\). Then the Galois group of \(W/\mathbb{Q}_p\) is cyclic of order \(n\), generated by the Frobenius automorphism \(\sigma\). The crossed product algebra of \(W\) and \(\text{gal}(W/\mathbb{Q}_p)\) leads to \(D\) by taking \(s\) such that \(s^n=p\) and the inner automorphism defined by \(s\) realizes \(\sigma\). The authors are concerned with the group of units in the maximal order in \(D\). It is a \(p\)-adic Lie group, called the Morava stabilizer group and denoted by \(S_n\). The subgroup of \(S_n\) consisting of strict units (i.e., \(u\) such that \(u\equiv 1\pmod s\)) is written \(S^0_n\) and the kernel of the reduced norm, restricted to \(S^0_n\) is denoted by \(S\ell\). Now the authors prove the following main theorem. \(S\ell\) contains a \(p\)th root of \(1\), \(\alpha\) such that (i) \(\{\alpha^{\omega i}\mid 1\leq i<p\}\) where \(\omega\) is a \((p-1)\)st root of \(-p\), generate an infinite subgroup \(G\) of \(S\ell\), isomorphic to the free product of \(p-1\) copies of \(\mathbb{Z}/p\), and (ii) \(G\) is dense in \(S\ell\). This result has applications to topology which are briefly discussed: each group \(S_n\) has a representation by automorphisms of a certain ring \(E_n\) and the cohomology of \(S_n\) with coefficients in this representation is related to the homotopy groups of the Bousfield localizations of the sphere, which in turn approximate the homotopy groups of spheres.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55N22
zbMATH DE Number: 1307796
zbMATH Keywords: division algebras
zbMATH Keywords: \(p\)-adic fields
zbMATH Keywords: index
zbMATH Keywords: Hasse invariants
zbMATH Keywords: Galois groups
zbMATH Keywords: crossed product algebras
zbMATH Keywords: groups of units
zbMATH Keywords: maximal orders
zbMATH Keywords: \(p\)-adic Lie groups
zbMATH Keywords: Morava stabilizer groups
zbMATH Keywords: strict units
zbMATH Keywords: reduced norms
zbMATH Keywords: free products
zbMATH Keywords: cohomology
zbMATH Keywords: homotopy groups
zbMATH Keywords: Bousfield localizations
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
full work available at URL: s0040-9383(97)00100-6
OpenAlex ID: W2048637021
Wikidata QID: Q128201179
description: scientific article; zbMATH DE number 1307796