Cœur et nombre d'intersection pour les actions de groupes sur les arbres
Cohomological dimension and symmetric automorphisms of a free group.
Combinatorial mapping-torus, branched surfaces and free group automorphisms
We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a -cocycle of a -complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction elucidates...
Commensurations of Out
Let Out(Fn) denote the outer automorphism group of the free group Fn with n>3. We prove that for any finite index subgroup Γ<Out(Fn), the group Aut(Γ) is isomorphic to the normalizer of Γ in Out(Fn). We prove that Γ is co-Hopfian: every injective homomorphism Γ→Γ is surjective. Finally, we prove that the abstract commensurator Comm(Out(Fn)) is isomorphic to Out(Fn).
Commutators and squares in free groups.
Conjugacy equivalence relation on subgroups
If G is a countable group containing a copy of F₂ then the conjugacy equivalence relation on subgroups of G attains the maximal possible complexity.
Co-rank and Betti number of a group
For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut number), and the maximal rank of its free abelian quotient, called the Betti number. We show that any combination of the group's rank, co-rank, and Betti number within obvious constraints is realized for some finitely presented group (for Betti number equal to rank, the group can be chosen torsion-free). In addition, we show that the Betti number is additive...
(Cyclic) subgroup separability of HNN and split extensions
Diophantine geometry over groups I : Makanin-Razborov diagrams
This paper is the first in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the first paper we present the (canonical) Makanin-Razborov diagram that encodes the set of solutions of a system of equations. We continue by studying parametric families of sets of solutions, and associate with such a family a canonical graded Makanin-Razborov diagram, that encodes the collection...
Efficient separability in free groups.
Elementary embeddings in torsion-free hyperbolic groups
We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if embeds elementarily in a torsion free hyperbolic group , we show that the group can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the...
Equations in groups, with special emphasis on localization and torsion - II
Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture.
Equivariant outer space and automorphisms of free-by-finite groups.
Erratum to: “Elementary embeddings in torsion-free hyperbolic groups”
Erratum to: Homology stability for outer automorphism groups of free groups.
Erratum: W. S. Jassim, On the intersection of finitely generated subgroups of free groups, Rev. Mat. Univ. Compl. 9(1996), 67-84.
Attention is drawn to an error in W. S. Jassim's paper.
Experimenting with infinite groups. I.
Fine convergence in free groups
Fixed points of endomorphisms of certain free products
The fixed point submonoid of an endomorphism of a free product of a free monoid and cyclic groups is proved to be rational using automata-theoretic techniques. Maslakova’s result on the computability of the fixed point subgroup of a free group automorphism is generalized to endomorphisms of free products of a free monoid and a free group which are automorphisms of the maximal subgroup.