Schreir graphs: Transitivity and coverings

Leemann, Paul-Henry

doi:10.1142/s021819671650003x

Cited by 6 publications

(6 citation statements)

References 11 publications

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“…Using Proposition 12 and Proposition 44 from [16], we obtain the following rigidity results about Cayley graphs of Tarski monsters. Both results are proved in Section 5.…”

Section: Tarski Monstersmentioning

confidence: 99%

“…If Γ is a graph, we say that a covering ϕ : Cay(G, T ) → Γ is compatible with the labels of Cay(G, S) if whenever two edges of Cay(G, S) have the same image under ϕ they have the same labels. Such coverings are in bijection with conjugacy classes of subgroups of G and that turns Γ into a Schreier graph of G, see [16]. Proposition 12.…”

Section: Coveringsmentioning

confidence: 99%

“…In particular, in [24], the second author together with Romain Tessera proved that if G is finitely presented and contains an element of infinite order, then there exist a finite generating set S and an integer R (depending only on the rank of G) such that every graph with the same balls of radius R as Cay(G, S) is covered by it. On the other hand, the first author showed in [16] that Cayley graphs of Tarski monsters do not cover other infinite transitive (even non simple) Schreier coset graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Recall that if Γ = (V, E) and ∆ = (W, F ) are graphs, a graph covering ϕ : Γ ∆ is a map V → W such that, for every v ∈ V , the restriction of ϕ to the set {v ∈ V | (v, v ) ∈ E} of neighbours of v is a bijection onto the set of neighbours of ϕ(v). We warn the reader that this definition is well adapted to our setting of simple graphs, but it has to be adapted if one wants to include also non simple graphs, see for example [16] for details. One way to construct covers of a Cayley graph Γ = Cay(G, S) is by taking the quotient by a subgroup H ≤ G (hence obtaining the so-called (right) Schreier coset graphs, which is a simple graph only under some conditions on H), but in general Γ may cover other graphs.…”

Section: Introductionmentioning

confidence: 99%

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Cayley graphs with few automorphisms

Leemann

Salle

2020

J Algebr Comb

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We show that every finitely generated group G with an element of order at least 5 rank (G) 12 admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above Cayley directed graph does not have bigons. On the other hand, if G is neither generalized dicyclic nor abelian and has an element of order at least (2 rank(G)) 36 , then it admits an undirected Cayley graph with automorphism group equal to G. This extends classical results for finite groups and free products of groups. The above results are obtained as corollaries of a stronger form of rigidity which says that the rigidity of the graph can be observed in a ball of radius 1 around a vertex. This strong rigidity result also implies that the Cayley (di)graph covers very few (di)graphs. In particular, we obtain Cayley graphs of Tarski monsters which essentially do not cover other quasi-transitive graphs. We also show that a finitely generated group admits a locally finite labelled unoriented Cayley graph with automorphism group equal to itself if and only if it is neither generalized dicyclic nor abelian with an element of order greater than 2.

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“…Using Proposition 12 and Proposition 44 from [16], we obtain the following rigidity results about Cayley graphs of Tarski monsters. Both results are proved in Section 5.…”

Section: Tarski Monstersmentioning

confidence: 99%

Section: Coveringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cayley graphs with few automorphisms

Leemann

Salle

2020

J Algebr Comb

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“…Graph theorists usually forbid the existence of degenerated loops, but these special kind of edges naturally arise in the context of Schreier graphs. Indeed, when allowing degenerated loops, there is a bijection between the subgroups of G (up to conjugacy) and graphs that are covered by Cayl(Γ; S), see [7] for a proof.…”

Section: Remarkmentioning

confidence: 99%

Up to a double cover, every regular connected graph is isomorphic to a Schreier graph

Leemann¹

2020

Preprint

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We prove that every connected locally finite regular graph has a double cover which is isomorphic to a Schreier graph.The aim of this short note is to provide a proof of the following result.Proposition 1. Let G be a d-regular connected graph. Then either G is isomorphic to a Schreier graph or G has a double-cover H which is isomorphic to a Schreier graph.While we were not able to find a reference to the above result in the literature, we do not claim any priority on it. In fact, this note was inspired by the following remark, which can be found at the end of Section 7 of [4]: "In fact up to the cover of degree 2 any regular graph can be realized as a Schreier graph [8]". However, [8] seems to treat only the case of graphs of even degree.Section 1 contains all the definitions as well as a discussion on the unusual concept of degenerated loop. The short Section 2 contains more materials on coverings and perfect matchings, as well as the proof of Proposition 1.

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On groups and simplicial complexes

Lubotzky

Luria

Rosenthal

2018

European Journal of Combinatorics

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The theory of k-regular graphs is closely related to group theory. Every k-regular, bipartite graph is a Schreier graph with respect to some group G, a set of generators S (depending only on k) and a subgroup H. The goal of this paper is to begin to develop such a framework for k-regular simplicial complexes of general dimension d. Our approach does not directly generalize the concept of a Schreier graph, but still presents an extensive family of k-regular simplicial complexes as quotients of one universal object: the k-regular d-dimensional arboreal complex, which is itself a simplicial complex originating in one specific group depending only on d and k. Along the way we answer a question from [PR16] on the spectral gap of higher dimensional Laplacians and prove a high dimensional analogue of Leighton's graph covering theorem. This approach also suggests a random model for k-regular d-dimensional multicomplexes.

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Schreir graphs: Transitivity and coverings

Cited by 6 publications

References 11 publications

Cayley graphs with few automorphisms

Cayley graphs with few automorphisms

Up to a double cover, every regular connected graph is isomorphic to a Schreier graph

On groups and simplicial complexes

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