scholarly journals Planar Transitive Graphs

10.37236/7888 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Matthias Hamann
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Transitive Graph ◽  
Locally Finite ◽  
Transitive Graphs ◽  
Finitely Presented

We prove that the first homology group of every planar locally finite transitive graph $G$ is finitely generated as an $\Aut(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible. 

Deficiency and commensurators

2018 ◽  
Vol 21 (3) ◽  
pp. 511-530
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Fundamental Group ◽  
Natural Homomorphism ◽  
Finitely Generated ◽  
Locally Finite ◽  
Finitely Generated Group ◽  
Locally Nilpotent

Abstract We show that if π is the fundamental group of a 4-dimensional infrasolvmanifold then {-2\leq\mathrm{def}(\pi)\leq 0} , and give examples realizing each value allowed by our constraints, for each possible value of the rank of {\pi/\pi^{\prime}} . We also consider the abstract commensurators of such groups. Finally, we show that if G is a finitely generated group, the kernel of the natural homomorphism from G to its abstract commensurator {\mathrm{Comm}(G)} is locally nilpotent by locally finite, and is finite if {\mathrm{def}(G)>1} .

A QUASI-ISOMETRY INVARIANT LOOP SHORTENING PROPERTY FOR GROUPS

2008 ◽  
Vol 18 (08) ◽  
pp. 1243-1257 ◽  
Author(s):  
STEPHEN G. BRICK ◽  
JON M. CORSON ◽  
DOHYOUNG RYANG
Keyword(s):  
Isoperimetric Inequality ◽  
Metric Spaces ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Quadratic Isoperimetric Inequality ◽  
Finitely Presented

We first introduce a loop shortening property for metric spaces, generalizing the property considered by M. Elder on Cayley graphs of finitely generated groups. Then using this metric property, we define a very broad loop shortening property for finitely generated groups. Our definition includes Elder's groups, and unlike his definition, our property is obviously a quasi-isometry invariant of the group. Furthermore, all finitely generated groups satisfying this general loop shortening property are also finitely presented and satisfy a quadratic isoperimetric inequality. Every CAT(0) cubical group is shown to have this general loop shortening property.

A survey on Hamiltonicity in Cayley graphs and digraphs on different groups

2019 ◽  
Vol 11 (05) ◽  
pp. 1930002
Author(s):  
G. H. J. Lanel ◽  
H. K. Pallage ◽  
J. K. Ratnayake ◽  
S. Thevasha ◽  
B. A. K. Welihinda
Keyword(s):  
Cayley Graphs ◽  
Current Status ◽  
Transitive Graph ◽  
Famous Conjecture ◽  
Hamilton Path ◽  
Vertex Transitive ◽  
Future Direction ◽  
Lovász Conjecture ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

Lovász had posed a question stating whether every connected, vertex-transitive graph has a Hamilton path in 1969. There is a growing interest in solving this longstanding problem and still it remains widely open. In fact, it was known that only five vertex-transitive graphs exist without a Hamiltonian cycle which do not belong to Cayley graphs. A Cayley graph is the subclass of vertex-transitive graph, and in view of the Lovász conjecture, the attention has focused more toward the Hamiltonicity of Cayley graphs. This survey will describe the current status of the search for Hamiltonian cycles and paths in Cayley graphs and digraphs on different groups, and discuss the future direction regarding famous conjecture.

On groups and 3-manifolds with weak dimension ≤ 1

1974 ◽  
Vol 75 (1) ◽  
pp. 33-35
Author(s):  
David E. Galewski
Keyword(s):  
Fundamental Group ◽  
Finitely Generated ◽  
Finite Generation ◽  
Connected Sum ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Weak Dimension ◽  
Finitely Presented

0. Introduction. A group π has weak dimension (wd) ≤ n (see Cartan and Ellen-berg (2)) if Hk(π, A) = 0 for all right Z(π)-modules A and all k > n. We say that the weak dimension of a manifold M is ≤ n if wd (πl(M))≤ n. In section 1 we show that open, orientable, irreducible 3-manifolds have wd ≤ 1 if and only if they are the monotone on of 1-handle bodies. In his celebrated theorem (10), Stallings proves that finitely presented groups of cohomological dimensions ≤ 1 are free. In section 2 we prove that if π is a finitely presented group which is the fundamental group of any orientable 3-manifold with wd ≤ 1 then π is free. We also give an example to show that the finite generation of π is necessary. (Swan (11) removes the finitely presented hypothesis from Stalling's theorem.) Finally, in section 3 we generalize a theorem of McMillan (5) and prove that if M is an open, orientable, irreducible 3-manifold with finitely generated fundamental group, then M is stably (taking the product with n ≥ 1 copies of ℝ) a connected sum along the boundary of trivial (n+2)-disc Sl bundles.

Bounded Depth Ascending HNN Extensions and -Semistability at infinity

2019 ◽  
Vol 72 (6) ◽  
pp. 1529-1550
Author(s):  
Michael L. Mihalik
Keyword(s):  
Fundamental Group ◽  
Companion Paper ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
New Methods ◽  
Hnn Extensions ◽  
Finitely Presented

AbstractA well-known conjecture is that all finitely presented groups have semistable fundamental groups at infinity. A class of groups whose members have not been shown to be semistable at infinity is the class ${\mathcal{A}}$ of finitely presented groups that are ascending HNN-extensions with finitely generated base. The class ${\mathcal{A}}$ naturally partitions into two non-empty subclasses, those that have “bounded” and “unbounded” depth. Using new methods introduced in a companion paper we show those of bounded depth have semistable fundamental group at infinity. Ascending HNN extensions produced by Ol’shanskii–Sapir and Grigorchuk (for other reasons), and once considered potential non-semistable examples are shown to have bounded depth. Finally, we devise a technique for producing explicit examples with unbounded depth. These examples are perhaps the best candidates to date in the search for a group with non-semistable fundamental group at infinity.

scholarly journals Self-Avoiding Walks and Amenability

10.37236/6577 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Geoffrey Grimmett ◽  
Zhongyang Li
Keyword(s):  
Growth Rate ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Transitive Graph ◽  
Amenable Groups ◽  
Height Functions ◽  
Connective Constant ◽  
Transitive Graphs ◽  
The Relationship ◽  
Self Avoiding Walks

The connective constant $\mu(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work.Various properties of connective constants depend on the existence of so-called 'unimodular graph height functions', namely: (i) whether $\mu(G)$ is a local function on certain graphs derived from $G$, (ii) the equality of $\mu(G)$ and the asymptotic growth rate of bridges, and (iii) whether there exists a terminating algorithm for approximating $\mu(G)$ to a given degree of accuracy.In the context of amenable groups, it is proved that the Cayley graphs of infinite, finitely generated, elementary amenable (and, more generally, virtually indicable) groups support unimodular graph height functions, which are in addition harmonic. In contrast, the Cayley graph of the Grigorchuk group, which is amenable but not elementary amenable, does not have a graph height function.In the context of non-amenable, transitive graphs, a lower bound is presented for the connective constant in terms of the spectral bottom of the graph. This is a strengthening of an earlier result of the same authors. Secondly, using a percolation inequality of Benjamini, Nachmias, and Peres, it is explained that the connective constant of a non-amenable, transitive graph with large girth is close to that of a regular tree. Examples are given of non-amenable groups without graph height functions, of which one is the Higman group.The emphasis of the work is upon the structure of Cayley graphs, rather than upon the algebraic properties of the underlying groups. New methods are needed since a Cayley graph generally possesses automorphisms beyond those arising through the action of the group.

scholarly journals Hopfian and co-Hopfian groups

1997 ◽  
Vol 56 (1) ◽  
pp. 17-24 ◽  
Author(s):  
Satya Deo ◽  
K. Varadarajan
Keyword(s):  
Fundamental Group ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Generated Groups ◽  
Aspherical Manifold ◽  
Finitely Presented

The main results proved in this note are the following:(i) Any finitely generated group can be expressed as a quotient of a finitely presented, centreless group which is simultaneously Hopfian and co-Hopfian.(ii) There is no functorial imbedding of groups (respectively finitely generated groups) into Hopfian groups.(iii) We prove a result which implies in particular that if the double orientable cover N of a closed non-orientable aspherical manifold M has a co-Hopfian fundamental group then π1(M) itself is co-Hopfian.

scholarly journals Vertex-transitive graphs which are not Cayley graphs, I

1994 ◽  
Vol 56 (1) ◽  
pp. 53-63 ◽  
Author(s):  
Brendan D. McKay ◽  
Cheryl E. Praeger
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Petersen Graph ◽  
Fourth Power ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractThe Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.

scholarly journals Cores of Vertex Transitive Graphs

10.37236/3144 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
David E. Roberson
Keyword(s):  
Cayley Graphs ◽  
Transitive Graph ◽  
The Core ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs ◽  
Vertex Sets ◽  
Normal Cayley Graphs

A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets each of which induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions.

scholarly journals Cayley Graphs Without a Bounded Eigenbasis

2020 ◽  
Author(s):  
Ashwin Sah ◽  
Mehtaab Sawhney ◽  
Yufei Zhao
Keyword(s):  
Cayley Graph ◽  
Orthonormal Basis ◽  
Cayley Graphs ◽  
The Other ◽  
Transitive Graph ◽  
Classic Result ◽  
Other Hand ◽  
Small Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graph

Abstract Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.

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