5,691 research outputs found
Group approximation in Cayley topology and coarse geometry, Part III: Geometric property (T)
In this series of papers, we study correspondence between the following: (1)
large scale structure of the metric space bigsqcup_m {Cay(G(m))} consisting of
Cayley graphs of finite groups with k generators; (2) structure of groups which
appear in the boundary of the set {G(m)}_m in the space of k-marked groups. In
this third part of the series, we show the correspondence among the metric
properties `geometric property (T),' `cohomological property (T),' and the
group property `Kazhdan's property (T).' Geometric property (T) of Willett--Yu
is stronger than being expander graphs. Cohomological property (T) is stronger
than geometric property (T) for general coarse spaces.Comment: 20 pages, Appendix withdrawn due to the error in the proof of Theorem
A.2 (v3); 24 pages, Appendix added (v2); 20 pages, no figur
The vertex-transitive TLF-planar graphs
We consider the class of the topologically locally finite (in short TLF)
planar vertex-transitive graphs, a class containing in particular all the
one-ended planar Cayley graphs and the normal transitive tilings. We
characterize these graphs with a finite local representation and a special kind
of finite state automaton named labeling scheme. As a result, we are able to
enumerate and describe all TLF-planar vertex-transitive graphs of any given
degree. Also, we are able decide to whether any TLF-planar transitive graph is
Cayley or not.Comment: Article : 23 pages, 15 figures Appendix : 13 pages, 72 figures
Submitted to Discrete Mathematics The appendix is accessible at
http://www.labri.fr/~renault/research/research.htm
Integral Cayley graphs and groups
We solve two open problems regarding the classification of certain classes of
Cayley graphs with integer eigenvalues. We first classify all finite groups
that have a "non-trivial" Cayley graph with integer eigenvalues, thus solving a
problem proposed by Abdollahi and Jazaeri. The notion of Cayley integral groups
was introduced by Klotz and Sander. These are groups for which every Cayley
graph has only integer eigenvalues. In the second part of the paper, all Cayley
integral groups are determined.Comment: Submitted June 18 to SIAM J. Discrete Mat
On Groupoids and Hypergraphs
We present a novel construction of finite groupoids whose Cayley graphs have
large girth even w.r.t. a discounted distance measure that contracts
arbitrarily long sequences of edges from the same colour class (sub-groupoid),
and only counts transitions between colour classes (cosets). These groupoids
are employed towards a generic construction method for finite hypergraphs that
realise specified overlap patterns and avoid small cyclic configurations. The
constructions are based on reduced products with groupoids generated by the
elementary local extension steps, and can be made to preserve the symmetries of
the given overlap pattern. In particular, we obtain highly symmetric, finite
hypergraph coverings without short cycles. The groupoids and their application
in reduced products are sufficiently generic to be applicable to other
constructions that are specified in terms of local glueing operations and
require global finite closure.Comment: Explicit completion of H in HxI (Section 2) is unstable (incompatible
with restrictions), hence does not support inductive construction towards
Prop. 2.17 based on Lem 2.16 as claimed. For corresponding technical result,
now see arxiv:1806.08664; for discussion of main applications first announced
here, now see arxiv:1709.0003
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