The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let \mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=\mathbf{G}(k_v). Let \Gamma be an arithmetic lattice in G and let C=C(\Gamma) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for \Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is \hat{F}_{\omega}, the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example \Gamma=SL_2(\mathcal{O}(S)), where \mathcal{O}(S) is the ring of S-integers in k, with S=\{v\}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of \Gamma on the Bruhat-Tits tree associated with G.Comment: 27 pages, 5 figures, to appear in J. Reine Angew. Mat

Spectral dependence of photoinduced spin precession in DyFeO3

Spin precession was nonthermally induced by an ultrashort laser pulse in orthoferrite DyFeO3 with a pump-probe technique. Both circularly and linearly polarized pulses led to spin precessions; these phenomena are interpreted as the inverse Faraday effect and the inverse Cotton-Mouton effect, respectively. For both cases, the same mode of spin precession was excited; the precession frequencies and polarization were the same, but the phases of oscillations were different. We have shown theoretically and experimentally that the analysis of phases can distinguish between these two mechanisms. We have demonstrated experimentally that in the visible region, the inverse Faraday effect was dominant, whereas the inverse Cotton-Mouton effect became relatively prominent in the near-infrared region.Comment: 27 pages, 8 figure