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

Geometry of generated groups with metrics induced by their Cayley color graphs

Let $G$ be a group and let $S$ be a generating set of $G$. In this article, we introduce a metric $d_C$ on $G$ with respect to $S$, called the cardinal metric. We then compare geometric structures of $(G, d_C)$ and $(G, d_W)$, where $d_W$ denotes the word metric. In particular, we prove that if $S$ is finite, then $(G, d_C)$ and $(G, d_W)$ are not quasi-isometric in the case when $(G, d_W)$ has infinite diameter and they are bi-Lipschitz equivalent otherwise. We also give an alternative description of cardinal metrics by using Cayley color graphs. It turns out that color-permuting and color-preserving automorphisms of Cayley digraphs are isometries with respect to cardinal metrics

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

Regular Embeddings of Canonical Double Coverings of Graphs

AbstractThis paper addresses the question of determining, for a given graphG, all regular maps havingGas their underlying graph, i.e., all embeddings ofGin closed surfaces exhibiting the highest possible symmetry. We show that ifGsatisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor productG⊗K2, can be described in terms of regular embeddings ofG. This allows us to “lift” the classification of regular embeddings of a given graph to a similar classification for its canonical double covering and to establish various properties of the “derived” maps by employing those of the “base” maps. We apply these results to determining all orientable regular embeddings of the tensor productsKn⊗K2(known as the cocktail-party graphs) and of then-dipolesDn, the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings ofKn⊗K2exist only ifnis a prime powerpl, and there are 2φ(n−1) orφ(n−1) isomorphism classes of such maps (whereφis Euler's function) according to whetherlis even or odd. Forleven an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings ofDnexist for each positive integern, and their number is a power of 2 depending on the decomposition ofninto primes