Thierry Coulbois

R-trees, dual laminations and compact systems of partial isometries

Let FN be a free group of finite rank N 2, and let T be an R-tree with a very small, minimal action of FN with dense orbits. For any basis A of FN there exists a heart KA ⊂ T (= the metric completion of T) which is a compact subtree that has the property that the dynamical system of partial isometries ai : KA ai KA → a −1 i KA KA, for each ai ∈ A, defines a tree T(KA,A) which contains an isometric copy of T as minimal subtree.

Equations in free groups are not finitely approximable

We give an equation in any free group F of rank > 2 that has a solution in any finite quotient of F, but has no solution in F.

FREE PRODUCT, PROFINITE TOPOLOGY AND FINITELY GENERATED SUBGROUPS

We consider the following property for a group G:(RZn)ifH1,…,Hnare finitely generated subgroups of G then the setH1 H2⋯ Hn= {h1 ⋯ hn| h1∈ H1, …,hn∈ Hn}is closed with respect to the profinite topology of G. It is obvious that finite groups and finitely generated commutative groups have the property (RZn). L. Ribes and P. Zalesskiĭ proved that any free group has (RZn). We show that the property (RZn) is stable under the free product operation. We use techniques developed by B. Herwig and D. Lascar on the one hand, R. Gitik on the other hand.

Ergodic currents dual to a real tree

Let

The class of groups which have a subgroup of index 2 is not elementary

T

Non-unique ergodicity, observers' topology and the dual algebraic lamination for

be an

\Bbb R

\R

-trees

-tree in the boundary of Outer space with dense orbits. When the free group

Rips induction: index of the dual lamination of an

\FN

\mathbb{R}

acts freely on