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Full-Text Articles in Physical Sciences and Mathematics
Infinitely Many Hyperbolic 3-Manifolds Which Contain No Reebless Foliation, R. Roberts, J. Shareshian, Melanie Stein
Infinitely Many Hyperbolic 3-Manifolds Which Contain No Reebless Foliation, R. Roberts, J. Shareshian, Melanie Stein
Faculty Scholarship
We investigate group actions on simply-connected (second countable but not necessarily Hausdorff) 1-manifolds and describe an infinite family of closed hyperbolic 3-manifolds whose fundamental groups do not act nontrivially on such 1-manifolds. As a corollary we conclude that these 3-manifolds contain no Reebless foliation. In fact, these arguments extend to actions on oriented -order trees and hence these 3-manifolds contain no transversely oriented essential lamination; in particular, they are non-Haken.
On Regular Graphs Optimally Labeled With A Condition At Distance Two, John P. Georges, David W. Mauro
On Regular Graphs Optimally Labeled With A Condition At Distance Two, John P. Georges, David W. Mauro
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For positive integers $j \geq k$, the $\lambda_{j,k}$-number of graph Gis the smallest span among all integer labelings of V(G) such that vertices at distance two receive labels which differ by at least k and adjacent vertices receive labels which differ by at least j. We prove that the $\lambda_{j,k}$-number of any r-regular graph is no less than the $\lambda_{j,k}$-number of the infinite r-regular tree $T_{\infty}(r)$. Defining an r-regular graph G to be $(j,k,r)$-optimal if and only if $\lambda_{j,k}(G) = \lambda_{j,k}(T_{\infty}(r))$, we establish the equivalence between $(j,k,r)$-optimal graphs and r-regular bipartite …