Search for: All records

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice $$X$$. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We show that the neighborhood of a minimal set is minimal. We characterize uniquely minimal sets of $$X$$: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of $$X$$: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph $$G$$ of minimal sets whose vertices are congruence classes of minimal sets of $$X$$ and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected. 

Abstract Sela proved that every torsion-free one-ended hyperbolic group is co-Hopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably co-Hopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably co-Hopfian.