Some basic properties of planar Singer groups

A planar Singer group is a collineation group of a finite (in this article) projective plane acting regularly on the points of the plane. Theorem 1 gives a characterization of abelian planar Singer groups. This leads to a necessary and sufficient condition for an inner automorphism to be a multiplier. The Sylow 2-structure of a multiplier group and some of its consequences are given in Theorem 3. One important result in studying multipliers of an abelian Singer group is the existence of a common fixed line. We extend this to an arbitrary planar Singer group in Theorem 4. Theorem 5 studies the order of an abelian group of multiplers. If this order equals to the order of the plane plus 1, then the number of points of the plane is a prime. If this order is odd, then it is at most the planar order plus 1.Partially supported by a NSA grant.     

Finite groups with planar subgroup lattices

It is natural to ask when a group has a planar Hasse lattice or more generally when its subgroup graph is planar. In this paper, we completely answer this question for finite groups. We analyze abelian groups, p-groups, solvable groups, and nonsolvable groups in turn. We find seven infinite families (four depending on two parameters, one on three, two on four), and three “sporadic” groups. In particular, we show that no nonabelian group whose order has three distinct prime factors can be planar.     

Characteristics of Graph Braid Groups

We give formulae for the first homology of the n-braid group and the pure 2-braid group over a finite graph in terms of graph-theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the n-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in Farber and Hanbury (arXiv:1005.2300 [math.AT]) can be verified. We discover more characteristics of graph braid groups: the n-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and the 2-braid group and the pure 2-braid group over a planar graph have a presentation whose relators are commutators. The latter was a conjecture in Farley and Sabalka (J. Pure Appl. Algebra, 2012) and so we propose a similar conjecture for higher braid indices.     

A complex for right-angled Coxeter groups

We associate to each right-angled Coxeter group a 2-dimensional complex. Using this complex, we show that if the presentation graph of the group is planar, then the group has a subgroup of finite index which is a 3-manifold group (that is, the group is virtually a 3-manifold group). We also give an example of a right-angled Coxeter group which is not virtually a 3-manifold group.

     

Planar embeddings with infinite faces

We investigate vertex‐transitive graphs that admit planar embeddings having infinite faces, i.e., faces whose boundary is a double ray. In the case of graphs with connectivity exactly 2, we present examples wherein no face is finite. In particular, the planar embeddings of the Cartesian product of the r‐valent tree with K₂ are comprehensively studied and enumerated, as are the automorphisms of the resulting maps, and it is shown for r = 3 that no vertex‐transitive group of graph automorphisms is extendable to a group of homeomorphisms of the plane. We present all known families of infinite, locally finite, vertex‐transitive graphs of connectivity 3 and an infinite family of 4‐connected graphs that admit planar embeddings wherein each vertex is incident with an infinite face. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 257–275, 2003     

Stable Planes from Lie Groups with Planar Partitions

We describe necessary and sufficient conditions under which a topological translation structure obtained from a Lie group with planar partition can be turned into a stable plane, thereby proving a conjecture due to Stroppel.     

Guionnet–Jones–Shlyakhtenko subfactors associated to finite-dimensional Kac algebras

We analyse the Guionnet–Jones–Shlyakhtenko construction for the planar algebra associated to a finite-dimensional Kac algebra and identify the factors that arise as finite interpolated free group factors.     

The planar algebra of group-type subfactors

If G is a countable, discrete group generated by two finite subgroups H and K and P is a II₁ factor with an outer G-action, one can construct the group-type subfactor PH⊂P?K introduced by Haagerup and the first author to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra of this subfactor and prove that any subfactor with an abstract planar algebra of “group type” arises from such a subfactor. The action of Jones' planar operad is determined explicitly.     

Embedding right-angled Artin groups into graph braid groups

We construct an embedding of any right-angled Artin group G(Δ) defined by a graph Δ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of Δ. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.      

On Spin Models,Triply Regular Association Schemes,and Duality

Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using series-parallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.     

The geometry at infinity of a hyperbolic Riemann surface of infinite type

We study geodesics on planar Riemann surfaces of infinite type having a single infinite end. Of particular interest is the class of geodesics that go out the infinite end in a most efficient manner. We investigate properties of these geodesics and relate them to the structure of the boundary of a Dirichlet polygon for a Fuchsian group representing the surface.      

Finite (line,plane)-flag-transitive planar spaces

We classify the finite planar spaces whose automorphism group acts transitively on the (line, plane)-flags.Dedicated to Professor Wagner on the occasion of his 60th birthday     

The dimension of planar posets

A partially ordered set (poset) is planar if it has a planar Hasse diagram. The dimension of a bounded planar poset is at most two. We show that the dimension of a planar poset having a greatest lower bound is at most three. We also construct four-dimensional planar posets, but no planar poset with dimension larger than four is known. A poset is called a tree if its Hasse diagram is a tree in the graph-theoretic sense. We show that the dimension of a tree is at most three and give a forbidden subposet characterization of two-dimensional trees.     

The Analytic and Formal Normal Form for the Nilpotent Singularity

We study orbital normal forms for analytic planar vector fields with nilpotent singularity. We show that the Takens normal form is analytic. In the case of generalized cusp we present the complete formal orbital normal form; it contains functional moduli. We interprete the coefficients of these moduli in terms of the hidden holonomy group.     

Martin Boundary of Killed Random Walks on Isoradial Graphs

Potential Analysis - We consider killed planar random walks on isoradial graphs. Contrary to the lattice case, isoradial graphs are not translation invariant, do not admit any group structure and...     

Quasiconformal groups and a theorem of Biship and Jones

We provide new bounds on the exponent of convergence of a planar discrete quasiconformal group in terms of the associated dilatation and the Hausdorff dimension of its conical limit set. In doing so, we use these bounds to realize a theorem of C. Bishop and P. Jones as an asymptotic limit in the dilatation.     

Isospectral polygons, planar graphs and heat content

Given a pair of planar isospectral, nonisometric polygons constructed as a quotient of the plane by a finite group, we construct an associated pair of planar isospectral, nonisometric weighted graphs. Using the natural heat operators on the weighted graphs, we associate to each graph a heat content. We prove that the coefficients in the small time asymptotic expansion of the heat content distinguish our isospectral pairs. As a corollary, we prove that the sequence of exit time moments for the natural Markov chains associated to each graph, averaged over starting points in the interior of the graph, provides a collection of invariants that distinguish isospectral pairs in general.

     

Further results on random cubic planar graphs

We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.