2,570 research outputs found
On planar Cayley graphs and Kleinian groups
Let be a finitely generated group acting faithfully and properly
discontinuously by homeomorphisms on a planar surface . We prove that admits such an action that is in addition
co-compact, provided we can replace by another surface .
We also prove that if a group has a finitely generated Cayley
(multi-)graph covariantly embeddable in , then can be
chosen so as to have no infinite path on the boundary of a face.
The proofs of these facts are intertwined, and the classes of groups they
define coincide. In the orientation-preserving case they are exactly the
(isomorphism types of) finitely generated Kleinian function groups. We
construct a finitely generated planar Cayley graph whose group is not in this
class.
In passing, we observe that the Freudenthal compactification of every planar
surface is homeomorphic to the sphere
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
Geodesics in Transitive Graphs
AbstractLetPbe a double ray in an infinite graphX, and letdanddPdenote the distance functions inXand inPrespectively. One callsPageodesicifd(x, y)=dP(x, y), for all verticesxandyinP. We give situations when every edge of a graph belongs to a geodesic or a half-geodesic. Furthermore, we show the existence of geodesics in infinite locally-finite transitive graphs with polynomial growth which are left invariant (set-wise) under “translating” automorphisms. As the main result, we show that an infinite, locally-finite, transitive, 1-ended graph with polynomial growth is planar if and only if the complement of every geodesic has exactly two infinite components
Entropy-Driven Phase Transition in Low-Temperature Antiferromagnetic Potts Models
We prove the existence of long-range order at sufficiently low temperatures, including zero temperature, for the three-state Potts antiferromagnet on a class of quasi-transitive plane quadrangulations, including the diced lattice. More precisely, we show the existence of (at least) three infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one sublattice have a higher probability to be in one state than in either of the other two states. For the special case of the diced lattice, we give a good rigorous lower bound on this probability, based on computer-assisted calculations that are not available for the other lattices
Behavior of Petrie Lines in Certain Edge-Transitive Graphs
We survey the construction and classification of one-, two- and infinitely-ended members of a class of highly symmetric, highly connected infinite graphs. In addition, we pose a conjecture concerning the relationship between the Petrie lines and ends of some infinitely-ended members of this class
On fixing boundary points of transitive hyperbolic graphs
We show that there is no 1-ended, planar, hyperbolic graph such that the
stabilizer of one of its hyperbolic boundary points acts transitively on the
vertices of the graph. This gives a partial answer to a question by Kaimanovich
and Woess.Comment: 9 page
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