Packing subgroups in solvable groups

We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska-Wise. In particular, the same is true about metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finitely generated subgroup without the bounded packing property. In this example the subgroup is a metabelian retract also. Thus we obtain a negative answer to Problem 2.27 of Hruska-Wise. On the other hand, we show that polycyclic subgroups of solvable groups satisfy the bounded packing property.Comment: 8 pages, no figur

Globalization and Rural Poverty

absolute poverty, self-employed, wage-employed, trade liberalization

Cosmic rays from remnants of quasars?

Considerations of the collision losses for protons traversing the 2.7 K black body microwave radiation field have led to the conclusion that the highest energy cosmic rays, those observed at $\geq 10^{20}$ eV, must come from sources within the present epoch. In light of this constraint, it is here suggested that these particles may be accelerated near the event horizons of spinning supermassive black holes associated with presently inactive quasar remnants. The required emf is generated by the black hole induced rotation of externally supplied magnetic field lines threading the horizon. Producing the observed flux of the highest energy cosmic rays would constitute a negligible drain on the black hole dynamo. Observations with upcoming air shower arrays and space missions may lead to the identification of candidate dormant galaxies which harbor such black holes. Although the highest energy events observed so far are accounted for within the context of this scenario, a spectral upper bound at $\sim 10^{21}$ eV is expected since the acceleration to higher energies appears to be precluded, on general grounds.Comment: Mon. Not. Roy. Astr. Soc., in press. 5 pages, 1 tabl