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Tullio Ceccherini-Silberstein

Publications and source records attributed to Tullio Ceccherini-Silberstein.

49 records · Page 3Linked to original sources
arXiv2011

On the density of periodic configurations in strongly irreducible subshifts

Let $G$ be a residually finite group and let $A$ be a finite set. We prove that if $X \subset A^G$ is a strongly irreducible subshift of finite type containing a periodic configuration then periodic configurations are dense in $X$. The density of periodic configurations implies in particular that every injective endomorphism of $X$ is surjective and that the group of automorphisms of $X$ is residually finite. We also introduce a class of subshifts $X \subset A^\Z$, including all strongly irreducible subshifts and all irreducible sofic subshifts, in which periodic configurations are dense.

math.DS↗
arXiv2011

On a family of Schreier graphs of intermediate growth associated with a self-similar group

For every infinite sequence $ω=x_1,x_2,...$, with $x_i\in\{0,1\}$, we construct an infinite 4-regular graph $X_ω$. These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space $\{0,1\}^{\infty}$. We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs $X_ω$ have intermediate growth.

math.GR↗
arXiv2011

On algebraic cellular automata

We investigate some general properties of algebraic cellular automata, i.e., cellular automata over groups whose alphabets are affine algebraic sets and which are locally defined by regular maps. When the ground field is assumed to be uncountable and algebraically closed, we prove that such cellular automata always have a closed image with respect to the prodiscrete topology on the space of configurations and that they are reversible as soon as they are bijective.

math.AG↗
arXiv2011

Expansive actions on uniform spaces and surjunctive maps

We present a uniform version of a result of M. Gromov on the surjunctivity of maps commuting with expansive group actions and discuss several applications. We prove in particular that for any group $Γ$ and any field $\K$, the space of $Γ$-marked groups $G$ such that the group algebra $\K[G]$ is stably finite is compact.

math.GR↗
arXiv2010

The Tutte Polynomial of the Schreier graphs of the Grigorchuk group and the Basilica group

We study the Tutte polynomial of two infinite families of finite graphs. These are the Schreier graphs associated with the action of two well-known self-similar groups acting on the binary rooted tree by automorphisms: the first Grigorchuk group of intermediate growth, and the iterated monodromy group of the complex polynomial $z^2-1$ known as the Basilica group. For both of them, we describe the Tutte polynomial and we compute several special evaluations of it, giving further information about the combinatorial structure of these graphs.

math.CO↗
arXiv2010

On the reversibility and the closed image property of linear cellular automata

When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space, it is known that every bijective linear cellular automaton $τ\colon V^G \to V^G$ is reversible and that the image of every linear cellular automaton $τ\colon V^G \to V^G$ is closed in $V^G$ for the prodiscrete topology. In this paper, we present a new proof of these two results which is based on the Mittag-Leffler lemma for projective sequences of sets. We also show that if $G$ is a non-periodic group and $V$ is an infinite-dimensional vector space, then there exist a linear cellular automaton $τ_1 \colon V^G \to V^G$ which is bijective but not reversible and a linear cellular automaton $τ_2 \colon V^G \to V^G$ whose image is not closed in $V^G$ for the prodiscrete topology.

math.GR↗
arXiv2010

A Garden of Eden theorem for linear subshifts

Let $G$ be an amenable group and let $V$ be a finite-dimensional vector space over an arbitrary field $\K$. We prove that if $X \subset V^G$ is a strongly irreducible linear subshift of finite type and $τ\colon X \to X$ is a linear cellular automaton, then $τ$ is surjective if and only if it is pre-injective. We also prove that if $G$ is countable and $X \subset V^G$ is a strongly irreducible linear subshift, then every injective linear cellular automaton $τ\colon X \to X$ is surjective.

math.DS↗
arXiv2009

Context-free pairs of groups I: Context-free pairs and graphs

Let $G$ be a finitely generated group, $A$ a finite set of generators and $K$ a subgroup of $G$. We call the pair $(G,K)$ context-free if the set of all words over $A$ that reduce in $G$ to an element of $K$ is a context-free language. When $K$ is trivial, $G$ itself is called context-free; context-free groups have been classified more than 20 years ago in celebrated work of Muller and Schupp as the virtually free groups. Here, we derive some basic properties of such group pairs. Context-freeness is independent of the choice of the generating set. It is preserved under finite index modifications of $G$ and finite index enlargements of $K$. If $G$ is virtually free and $K$ is finitely generated then $(G,K)$ is context-free. A basic tool is the following: $(G,K)$ is context-free if and only if the Schreier graph of $(G,K)$ with respect to $A$ is a context-free graph.

math.GR↗