Generalizations of the Muller-Schupp theorem and tree-like inverse graphs

We extend the characterization of context-free groups of Muller and Schupp in two ways. We first show that for a quasi-transitive inverse graph $\Gamma$, being quasi-isometric to a tree, or context-free (finitely many end-cones types), or having the automorphism group $Aut(\Gamma)$ that is virtually free, are all equivalent conditions. Furthermore, we add to the previous equivalences a group theoretic analog to the representation theorem of Chomsky-Sch\"utzenberger that is fundamental in solving a weaker version of a conjecture of T. Brough which also extends Muller and Schupp' result to the class of groups that are virtually finitely generated subgroups of direct product of free groups. We show that such groups are precisely those whose word problem is the intersection of a finite number of languages accepted by quasi-transitive, tree-like inverse graphs

Homogeneity in the free group

We show that any non abelian free group \F is strongly $\aleph_0$-homogeneous, i.e. that finite tuples of elements which satisfy the same first-order properties are in the same orbit under \Aut(\F). We give a characterization of elements in finitely generated groups which have the same first-order properties as a primitive element of the free group. We deduce as a consequence that most hyperbolic surface groups are not $\aleph_0$-homogeneous.Comment: 26 page

Some relational structures with polynomial growth and their associated algebras II: Finite generation

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $A(R)$, introduced by P.~J.~Cameron. In a previous paper, we studied the relationship between the properties of a relational structure and those of their algebra, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. In this paper, we investigate how far the well know algebraic properties of those rings extend to age algebras. The main result is a combinatorial characterization of when the age algebra is finitely generated. In the special case of tournaments, we show that the age algebra is finitely generated if and only if the profile is bounded. We explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra.Comment: 27 pages; submitte