Infinite subgroups of the Morava stabilizer groups
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.
scientific article; zbMATH DE number 1307796