A characterization of virtually embedded subsurfaces in 3-manifolds

The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this case, the subsurface is virtually a leaf of a taut foliation. Besides other consequences, examples are exhibited that non-geometric 3-manifolds with no Seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces.Comment: 28 pages. Errors of previous Proposition 3.1 and Formula 7.2 correcte

A Jorgensen-Thurston theorem for homomorphisms

In this note, we provide a description of the structure of homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jorgensen-Thurston Theorem in hyperbolic geometry.Comment: 16 pages, 4 figure

Optimal Relay Selection with Non-negligible Probing Time

In this paper an optimal relay selection algorithm with non-negligible probing time is proposed and analyzed for cooperative wireless networks. Relay selection has been introduced to solve the degraded bandwidth efficiency problem in cooperative communication. Yet complete information of relay channels often remain unavailable for complex networks which renders the optimal selection strategies impossible for transmission source without probing the relay channels. Particularly when the number of relay candidate is large, even though probing all relay channels guarantees the finding of the best relays at any time instant, the degradation of bandwidth efficiency due to non-negligible probing times, which was often neglected in past literature, is also significant. In this work, a stopping rule based relay selection strategy is determined for the source node to decide when to stop the probing process and choose one of the probed relays to cooperate with under wireless channels' stochastic uncertainties. This relay selection strategy is further shown to have a simple threshold structure. At the meantime, full diversity order and high bandwidth efficiency can be achieved simultaneously. Both analytical and simulation results are provided to verify the claims.Comment: 8 pages. ICC 201