We study the infinite family of spider-web graphs {S k,N,M }, k ≥ 2, N ≥ 0 and M ≥ 1, initiated in the 50-s in the context of network theory. It was later shown in physical literature that these graphs have remarkable percolation and spectral properties. We provide a mathematical explanation of these properties by putting the spider-web graphs in the context of group theory and algebraic graph theory. Namely, we realize them as tensor products of the well-known de Bruijn graphs {B k,N } with cyclic graphs {CM } and show that these graphs are described by the action of the lamplighter group L k = Z/kZ Z on the infinite binary tree. Our main result is the identification of the infinite limit of {S k,N,M }, as N, M → ∞, with the Cayley graph of the lamplighter group L k which, in turn, is one of the famous Diestel-Leader graphs DL k,k . As an application we compute the spectra of all spider-web graphs and show their convergence to the discrete spectral distribution associated with the Laplacian on the lamplighter group.

Let $G$ be a finitely generated regular branch group acting by automorphisms on a regular rooted tree $T$. It is well-known that stabilizers of infinite rays in $T$ (aka parabolic subgroups) are weakly maximal subgroups in $G$, that is, maximal among subgroups of infinite index. We show that, given a finite subgroup $Q\leq G$, $G$ possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing $Q$. In particular, for Grigorchuk-Gupta-Sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups that are not parabolic.Comment: 11 pages; a few corrections; to appear in J. Algebr

We give a characterization of isomorphisms between Schreier graphs in terms of the groups, subgroups and generating systems. This characterization may be thought as a graph analog of Mostow's rigidity theorem for hyperbolic manifolds. This allows us to give a transitivity criterion for Schreier graphs. Finally, we show that Tarski monsters satisfy a strong simplicity criterion.

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.

We characterize the finitely generated groups that admit a Cayley graph whose only automorphisms are the translations, confirming a conjecture by Watkins from 1976. The proof relies on random walk techniques. As a consequence, every finitely generated group admits a Cayley graph with countable automorphism group. We also treat the case of directed graphs.Résumé. -Nous caractérisons les groupes de type fini qui admettent un graphe de Cayley dont les seuls automorphismes sont les translations. Cela confirme une conjecture de Watkins