Related papers: Strongly transitive multiple trees

We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability…

For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group,…

We prove that given a fixed finite tree $P$, almost all trees contain $P$ as a subtree. Moreover, the inclusion can be made so that it induces an embedding of the corresponding (quantum) automorphism groups, thereby providing generic…

An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k \geq 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if…

This work addresses the existence of transitive extensions of certain infinite permutation groups which arise as the automorphism groups of model-theoretic structures which are generic in the Fra\"iss\'e sense. The study of transitive…

We consider actions of completely metrisable groups on simplicial trees in the context of the Bass--Serre theory. Our main result characterises continuity of the amplitude function corresponding to a given action. Under fairly mild…

Let $T$ be a locally finite tree all of whose vertices have valency at least $6$. We classify, up to isomorphism, the closed subgroups of $\mathrm{Aut}(T)$ acting $2$-transitively on the set of ends of $T$ and whose local action at each…

Let T be a d-regular tree (d > 2) and A=Aut(T), its automorphism group. Let G be a group generated by n independent Haar-random elements of A. We show that almost surely, every nontrivial element of G has finitely many fixed points on T.

We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free…

For a finite alphabet $\mathcal{A}$ and shift $X\subseteq\mathcal{A}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group ${\rm Aut}(X)$. For such systems, we…

We show that smooth well formed weighted complete intersections have finite automorphism groups, with several obvious exceptions.

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to…

We characterize free products admitting a faithful and highly transitive action. In particular, we show that the group $\PSL_2(\Z)\simeq (\Z/2\Z)*(\Z/3\Z)$ admits a faithful and highly transitive action on a countable set.

We study hereditary properties of the class of countable groups admitting an amenable, transitive and faithful action on a countable set. We consider mainly the case of amalgamated free products, and we show in particular that the double of…

We show that the automorphism group of a certain subclass of smooth Gizatullin surfaces with a distinguished and rigid extended divisor is generated by automorphisms of A1-fibrations. Moreover, such surfaces provide examples of smooth…

The group A_{m} of automophisms of a one-rooted m-ary tree admits a diagonal monomorphism which we denote by x. Let A be an abelian state-closed (or self-similar) subgroup of A_{m}. We prove that the combined diagonal and tree-topological…

We investigate the connections between tree amalgamations and quasi-isometries. In particular, we prove that the quasi-isometry type of multi-ended accessible quasi-transitive connected locally finite graphs is determined by the…

We realize several combinatorial Hopf algebras based on set compositions, plane trees and segmented compositions in terms of noncommutative polynomials in infinitely many variables. For each of them, we describe a trialgebra structure, an…

We study random trees which are invariant in law under the operation of contracting each edge independently with probability $p\in(0,1)$. We show that all such trees can be constructed through Poissonian sampling from a certain class of…

We prove the existence of an automorphism-invariant coupling for the wired and the free uniform spanning forests on Cayley graphs of finitely generated residually amenable groups.