Related papers: Cayley graphs with few automorphisms

We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such that every such automorphism of every connected Cayley graph on G has a very simple form: the…

We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to…

We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex…

We determine two new infinite families of Cayley graphs that admit colour-preserving automorphisms that do not come from the group action. By definition, this means that these Cayley graphs fail to have the CCA (Cayley Colour Automorphism)…

We define the notion of rough Cayley graph for compactly generated locally compact groups in terms of quasi-actions. We construct such a graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to…

The characterization of distance-regular Cayley graphs originated from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a classification of distance-regular Cayley…

For a finite group $G$ and an inverse-closed generating set $C$ of $G$, let $Aut(G;C)$ consist of those automorphisms of $G$ which leave $C$ invariant. We define an $Aut(G;C)$-invariant normal subgroup $\Phi(G;C)$ of $G$ which has the…

We give a computer-assisted proof that if $G$ is a finite group of order $8pq$, where $p$ and $q$ are distinct primes, then every connected Cayley graph on $G$ has a hamiltonian cycle.

We study those automatic sequences which are produced by an automaton whose underlying graph is the Cayley graph of a finite group. For $2$-automatic sequences, we find a characterization in terms of what we call homogeneity, and among…

Let $G$ be a group and $m$ a positive integer. We say an $m$-Cayley digraph $\Gamma$ over $G$ is a digraph that admits a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. The digraph $\Sigma$…

We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in…

A result of Pyber states that every finite group $G$ contains an abelian subgroup whose order is quasi-polynomially large in $\lvert G\rvert$. We prove a similar result for $K$-approximate subgroups of solvable groups under only modest…

We prove that random d-regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (log_{d-1}|G|)^{1/2}/2 and that random d-regular Cayley graphs of simple algebraic groups over F_q asymptotically almost…

The girth of a finitely generated group G is the supremum of the girth of Cayley graphs for G over all finite generating sets. Let G be a finitely generated subgroup of the mapping class group Mod(S), where S is a compact orientable…

In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group $D_n=\langle a,b\mid a^n=b^2=1,bab=a^{-1}\rangle$. Moreover, we also obtain some simple sufficient conditions for the…

We give an introduction to the Cayley-Abels graph for a totally disconnected, locally compact (tdlc) group. It is a generalization of the Cayley graph. We illustrate that on the one hand, Cayley-Abels graphs are useful tools to extend…

We bound the volume of thick embeddings of finite graphs into the Heisenberg group, as well as the volume of coarse wirings of finite graphs into groups with polynomial growth. This work follows the work of Kolmogorov-Brazdin, Gromov-Guth…

It is shown that a flat subgroup, $H$, of the totally disconnected, locally compact group $G$ decomposes into a finite number of subsemigroups on which the scale function is multiplicative. The image, $P$, of a multiplicative semigroup in…

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a…

A discrete analog of quantum unique ergodicity was proved for Cayley graphs of quasirandom groups by Magee, Thomas and Zhao. They show that for large graphs there exist real orthonormal basis of eigenfunctions of the adjacency matrix such…