2,350 research outputs found
The planar Cayley graphs are effectively enumerable I: consistently planar graphs
We obtain an effective enumeration of the family of finitely generated groups
admitting a faithful, properly discontinuous action on some 2-manifold
contained in the sphere. This is achieved by introducing a type of group
presentation capturing exactly these groups.
Extending this in a companion paper, we find group presentations capturing
the planar finitely generated Cayley graphs. Thus we obtain an effective
enumeration of these Cayley graphs, yielding in particular an affirmative
answer to a question of Droms et al.Comment: To appear in Combinatorica. The second half of the previous version
is arXiv:1901.0034
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
Self-avoiding walks and connective constants
The connective constant of a quasi-transitive graph is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph .
We present upper and lower bounds for in terms of the
vertex-degree and girth of a transitive graph.
We discuss the question of whether for transitive
cubic graphs (where denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
We present strict inequalities for the connective constants
of transitive graphs , as varies.
As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with
arXiv:1304.721
On the Number of Embeddings of Minimally Rigid Graphs
Rigid frameworks in some Euclidian space are embedded graphs having a unique
local realization (up to Euclidian motions) for the given edge lengths,
although globally they may have several. We study the number of distinct planar
embeddings of minimally rigid graphs with vertices. We show that, modulo
planar rigid motions, this number is at most . We also exhibit several families which realize lower bounds of the order
of , and .
For the upper bound we use techniques from complex algebraic geometry, based
on the (projective) Cayley-Menger variety over the complex numbers . In this context, point configurations
are represented by coordinates given by squared distances between all pairs of
points. Sectioning the variety with hyperplanes yields at most
zero-dimensional components, and one finds this degree to be
. The lower bounds are related to inductive
constructions of minimally rigid graphs via Henneberg sequences.
The same approach works in higher dimensions. In particular we show that it
leads to an upper bound of for the number of spatial embeddings
with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to
rigid motions
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