Related papers: Groups that produce expander graphs
It is well known there is no finitely generated abelian group which has the $R_\infty$ property. We will show that also many non-finitely generated abelian groups do not have the $R_\infty$ property, but this does not hold for all of them.…
When one studies geometric properties of graphs, local finiteness is a common implicit assumption, and that of transitivity a frequent explicit one. By compactness arguments, local finiteness guarantees several regularity properties. It is…
We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. $>\frac{1}{2}$) but strictly smaller than any lattice (i.e. $<1$). More precisely, every affine covering of a…
We prove that there exist $k\in N$ and $0<\epsilon\in R$ such that every non-abelian finite simple group $G$, which is not a Suzuki group, has a set of $k$ generators for which the Cayley graph $\Cay(G; S)$ is an $\epsilon$-expander.
Let $S$ be a subset of a amenable group $G$ such that $e\in S$ and $S^{-1}=S$. The main result of the paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite…
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…
We give a characterization of the covering Schreier graphs of groups generated by bounded automata to be Galois. We also investigate the zeta and L functions of Schreier graphs of few groups namely the Grigorchuk group, Gupta-Sidki p group,…
We prove that if G is SL_2(F) or PSL_2(F), where F is a finite field, and A is a set of generators of G, then either |AAA| > |A|^(1+epsilon), where epsilon is an absolute positive real number, or AAA=G. As a corollary we get that the…
We prove that the boundary action of a sofic random subgroup of a finitely generated free group is conservative. This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda, who studied the boundary actions of individual…
The unipotent subgroup of a finite group of Lie type over a prime field Z/pZ comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present…
It is known that splittings of finitely presented groups over 2-ended groups can be characterized geometrically. We show that this characterization does not extend to all finitely generated groups. Answering a question of Kleiner we show…
The cogrowth series of a group with respect to a finite generating set is an important combinatorial quantity that seems very difficult to compute exactly, as evidenced by the scarcity of known examples. In this paper, we give a particular…
We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of…
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…
Every automaton group naturally acts on the space $X^\omega$ of infinite sequences over some alphabet $X$. For every $w\in X^\omega$ we consider the Schreier graph $\Gamma_w$ of the action of the group on the orbit of $w$. We prove that for…
We construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of $SL_2 (\mathbb{F}_p [t])$ modulo certain square-free ideals. We describe some applications of our results…
A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on…
In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is a family of expanders under certain algebraic conditions. Here is…
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…
In the present paper, we study Neumaier Cayley graphs. First, we give a criterion for a Cayley graph to be a Neumaier graph with a spread given by the cosets of a subgroup. Further, we construct a new infinite family of Neumaier Cayley…