We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.
As a main result of this paper we give conditions under which the generalized $X$-join of Cayley graphs is a Cayley graph. In particular, we show that $X$-join of isomorphic Cayley graphs is a Cayley grpah. To do this, new properties for a generalized wreath product of permutation groups are given in the case where the base group acts regularly. These are used to give conditions for the generalized wreath product to contain a regular subgroup, which are then applied to generalized $X$-joins of Cayley graphs. Along the way, it is shown that the generalized $X$-join of isomorphic Cayley graphs w…
We show that for certain integers $n$, the problem of whether or not a Cayley digraph $Γ$ of $\mathbb Z_n$ is also isomorphic to a Cayley digraph of some other abelian group $G$ of order $n$ reduces to the question of whether or not a natural subgroup of the full automorphism group contains more than one regular abelian group up to isomorphism (as opposed to the full automorphism group). A necessary and sufficient condition is then given for such circulants to be isomorphic to Cayley digraphs of more than one abelian group, and an easy-to-check necessary condition is provided.
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.
We prove that every vertex transitive, planar, 1-ended, graph covers every graph whose balls of radius r are isomorphic to the ball of radius r in G for a sufficiently large r. We ask whether this is a general property of finitely presented Cayley graphs, as well as further related questions.
A Cayley graph of a group $H$ is a finite simple graph $Γ$ such that ${\rm Aut}(Γ)$ contains a subgroup isomorphic to $H$ acting regularly on $V(Γ)$, while a Haar graph of $H$ is a finite simple bipartite graph $Σ$ such that ${\rm Aut}(Σ)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(Σ)$ and the $H$-orbits are equal to the bipartite sets of $Σ$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner…
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 concepts concerning finitely generated groups to compactly generated, tdlc groups and on the other hand, they can be used to investigate properties that, for finitely generated groups, are trivial.
A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_4\times C_p^2$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$.
We show that a group admits a planar, finitely generated Cayley graph if and only if it admits a special kind of group presentation we introduce, called a planar presentation. Planar presentations can be recognised algorithmically. As a consequence, we obtain an effective enumeration of the planar Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al. asking whether the planar groups can be effectively enumerated.
A diagonal base of a Sylow 2-subgroup $P_n(2)$ of symmetric group $S_{2^n}$ is a minimal generating set of this subgroup consisting of elements with only one non-zero coordinate in the polynomial representation. For different diagonal bases Cayley graphs of $P_n(2)$ may have different girths (i.e. minimal lengths of cycles) and thus be non-isomorphic. In presented paper all possible values of girths of Cayley graphs of $P_n(2)$ on diagonal bases are calculated. A criterion for whenever such Cayley graph has girth equal to 4 is presented. A lower bound for the number of different non-isomorphic…
The aim of this small note is to prove an elementary yet useful properties of finitely presented groups. Let G be a finitely generated group with one end. Fix a (finite) generating set and let $B_n$ be the ball of radius $n$ around $e$. Let $B_n^{c,\infty}$ be the infinite connected component of the complement of $B_n$. Then G has connected spheres if there exists a $r >0$ such that $B_{n+r} \cap B_n^{c,\infty}$ is connected for all $n \geq 0$. This note shows that if G is finitely presented then it has connected spheres.
A finite group $G$ admits an {\em oriented regular representation} if there exists a Cayley digraph of $G$ such that it has no digons and its automorphism group is isomorphic to $G$. Let $m$ be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented $m$-semiregular representations using $m$-Cayley digraphs. Given a finite group $G$, an {\em $m$-Cayley digraph} of $G$ is a digraph that has a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. We say that a finite group $G$ admits an {\em oriented $m$-se…
A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_2^5\times C_p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$. Together with the previously obtained results, this implies that a group $G$ of order $32p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$ and $G\cong C_2^5\times C_p$.
We introduce and study a strong "thin triangle"' condition for directed graphs, which generalises the usual notion of hyperbolicity for a metric space. We prove that finitely generated left cancellative monoids whose right Cayley graphs satisfy this condition must be finitely presented with polynomial Dehn functions, and hence word problems in NP. Under the additional assumption of right cancellativity (or in some cases the weaker condition of bounded indegree), they also admit algorithms for more fundamentally semigroup-theoretic decision problems such as Green's relations L, R, J, D and the …
We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory, including: the 1-skeletons of systolic and quadric complexes (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), 1-skeletons of finite dimensional CAT(0) cube complexes, hyperbolic graphs, standard Cayley graphs of finitely generated…
In this work we characterise Cayley graphs of Coxeter groups with respect to the standard generating set that admit uncountable vertex stabilisers. As a corollary, we fully identify finitely generated Coxeter groups for which the automorphism group of their Cayley graph with respect to the standard generating set is not discrete when equipped with the permutation topology. As an application, we also provide new explicit constructions of vertex-transitive graphs of infinite degree that have locally compact automorphism groups.
Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $Γ$, does there exist a generating set $S$ such that the Cayley graph $(Γ,S)$, without loops and multiple edges, has non-unique percolation, i.e., $p_c(Γ,S)<p_u(Γ,S)$? We show that this is true if $Γ$ contains an infinite normal subgroup $N$ such that $Γ/ N$ is non-amenable. Moreover for any finitely generated group $G$ containing $Γ$ there exists a generating set $S'$ of $G$ such that $p_c(G,S')<p_u(G,S')$. In particular this applies to fr…
We prove that the first homology group of every planar locally transitive finite graph $G$ is a finitely generated ${\rm Aut}(G)$-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.
In this paper, we characterize the finite groups $G$ of even order with the property that for any involution $x$ and element $y$ of $G$, $\langle x, y \rangle$ is isomorphic to one of the following groups: $\mathbb{Z}_2,$ $\mathbb{Z}_2^2$, $\mathbb{Z}_4$, $\mathbb{Z}_6$, $\mathbb{Z}_2 \times \mathbb{Z}_4$, $\mathbb{Z}_2 \times\mathbb{Z}_6$ and $A_4$. As a result, a characterization will be obtained for the finite groups all of whose Cayley graphs of degree $3$ have integral spectrum.
A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to G. We prove that up to finite index, these are always obtained by pulling back a chain of subgroups from a free abelian quotient. We give two applications: First, we show any characteristic proper finite-index subgroup isomorphic to G arises by pulling back a finite-index subgroup of the abelianization, and secondly, we prove special cases (fo…
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 $Φ(G;C)$ of $G$ which has the property that, for any $Aut(G;C)$-invariant normal set of generators for $G$, if we remove from it all the elements of $Φ(G;C)$, then the remaining set is still an $Aut(G;C)$-invariant normal generating set for $G$. The subgroup $Φ(G;C)$ contains the Frattini subgroup $Φ(G)$ but the inclusion may be proper. The Cayley graph $Cay(G,C)$ is normal edge-transitive if $A…
We introduce right generating sets, Cayley graphs, growth functions, types and rates, and isoperimetric constants for left homogeneous spaces equipped with coordinate systems; characterise right amenable finitely right generated left homogeneous spaces with finite stabilisers as those whose isoperimetric constant is $0$; and prove that finitely right generated left homogeneous spaces with finite stabilisers of sub-exponential growth are right amenable, in particular, quotient sets of groups of sub-exponential growth by finite subgroups are right amenable.
In contrast to being automatic, being Cayley automatic \emph{a priori} has no geometric consequences. Specifically, Cayley graphs of automatic groups enjoy a fellow traveler property. Here we study a distance function introduced by the first author and Trakuldit which aims to measure how far a Cayley automatic group is from being automatic, in terms of how badly the Cayley graph fails the fellow traveler property. The first author and Trakuldit showed that if it fails by at most a constant amount, then the group is in fact automatic. In this article we show that for a large class of non-automa…
A one-by-one exhaustion is a combinatorial/geometric condition which excludes eigenvalues from the spectra of Laplace and Schrödinger operators on graphs. Isoperimetric inequalities in graphs with a cocompact automorphism group provide an upper bound on the von Neumann dimension of the space of eigenfunctions. Any finitely generated indicable amenable group has a Cayley graph without eigenvalues. There exists a finitely generated group G with finite generating sets S and S' such that the adjacency operator of the Cayley graph of (G,S) has no eigenvalue while the adjacency operator of the Cayle…
The pro-isomorphic zeta function of a torsion-free finitely generated nilpotent group G enumerates finite index subgroups H such that H and G have isomorphic profinite completions. It admits an Euler product decomposition, indexed by the rational primes. We manufacture the first example of a torsion-free finitely generated nilpotent group G such that the local Euler factors of its pro-isomorphic zeta function do not satisfy functional equations. The group G has nilpotency class 4 and Hirsch length 25. It is obtained, via the Malcev correspondence, from a Z-Lie lattice L with a suitable algebra…
It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we exhibit various examples of Cayley graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgako…
The connective constant $μ(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called 'unimodular graph height functions'. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph …
We construct the first example of a finitely-presented, residually-finite group that contains an infinite sequence of non-isomorphic finitely-presented subgroups such that each of the inclusion maps induces an isomorphism of profinite completions.
