Vincent Guirardel

Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups,

Limit groups as limits of free groups: compactifying the set of free groups.

Out(F_n)

Limit groups and groups acting freely on R n -trees

, and the Cremona group. Other examples can be found among groups acting geometrically on

Cœur et nombre d'intersection pour les actions de groupes sur les arbres

CAT(0)

Dynamics of Out(Fn) on the boundary of outer space

spaces, fundamental groups of graphs of groups, etc. We obtain a number of general results about rotating families and hyperbolically embedded subgroups; although our technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, we solve two open problems about mapping class groups, and obtain some results which are new even for relatively hyperbolic groups.

Deformation spaces of trees

We give a topological framework for the study of Selaslimit groups: limit groups are limits of free groups in a compact space of marked groups. Many results get a natural interpretation in this setting. The class of limit groups is known to coincide with the class of finitely generated fully residually free groups. The topological approach gives some new insight on the relation between fully residually free groups, the universal theory of free groups, ultraproducts and non-standard free groups.

Foliations for solving equations in groups: free, virtually free, and hyperbolic groups

We give a simple proof of the finite presentation of Sela’s limit groups by using free actions on R n –trees. We first prove that Sela’s limit groups do have a free action on an R n –tree. We then prove that a finitely generated group having a free action on an R n –tree can be obtained from free abelian groups and surface groups by a finite sequence of free products and amalgamations over cyclic groups. As a corollary, such a group is finitely presented, has a finite classifying space, its abelian subgroups are finitely generated and contains only finitely many conjugacy classes of non-cyclic maximal abelian subgroups.

Trees of cylinders and canonical splittings

We present the construction of some kind of convex core for the product of two actions of a group on R-trees. This geometric construction allows to generalize and unify the intersection number of two curves or of two measured foliations on a surface, Scotts intersection number of splittings, and the apparition of surfaces in Fujiwara-Papasoglus construction of the JSJ splitting. In particular, this construction allows a topological interpretation of the intersection number analogous to the definition for curves in surfaces. As an application of this construction, we prove that an irreducible automorphism of the free group whose stable and unstable trees are geometric, is actually induced a pseudo-Anosov homeomorphism on a surface. Consider a surface Σ and two homotopy classes of simple closed curves c 1 , c 2 ⊂ Σ. Denote by i(c 1 , c 2) their geometric intersection number. The nullity of the intersection number is equivalent to the possibility of isotoping the curves apart. In terms of splittings, i(c 1 , c 2) = 0 if and only if the two splittings of π 1 (Σ) dual to these curves are compatible (i. e. have a common refinement). In [Sco98] (see also [SS00, SS03]) Scott generalized this notion of intersection number to any pair of splittings of a finitely generated group G. This intersection number is always symmetric. Moreover, when edge groups of the splittings are finitely generated then this intersection number is finite, and it vanishes if and only if the two splittings are compatible. By Bass-Serre theory, two splittings of a group G correspond to two actions of G on simplicial trees T 1 , T 2. In this article, we give a geometric construction of a kind of convex core for the diagonal action of G on T 1 × T 2 which captures the information about the intersection number of the 1 corresponding splittings. Because of its geometric nature, this construction works naturally in the context of R-trees. The convexity in question here is not the CAT(0) convexity, which would give a much too large set. The useful notion in this context is fiberwise convexity: a subset E ⊂ T 1 × T 2 has convex fibers if for both i ∈ {1, 2} and every x ∈ T i , E ∩ p −1 i (x) is convex (where p i : T 1 × T 2 → T i denotes …

Splittings and automorphisms of relatively hyperbolic groups

In this paper, we study the dynamics of the action of Out(Fn) on the boundary ∂CVn of outer space: we describe a proper closed Out(Fn)-invariant subset Fn of ∂CVn such that Out(Fn) acts properly discontinuously on the complementary open set. Moreover, we prove that there is precisely one minimal non-empty closed invariant subset Mn in Fn. This set Mn is the closure of the Out(Fn)-orbit of any simplicial action lying in Fn. We also prove that Mn contains every action having at most n−1 ergodic measures. This makes us suspect that Mn=Fn. Thus Fn would be the limit set of Out(Fn), the complement of Fn being its set of discontinuity.

Presenting parabolic subgroups

Let G be a finitely generated group. Two simplicial G-trees are said to be in the same deformation space if they have the same elliptic subgroups (if H fixes a point in one tree, it also does in the other). Examples include Culler-Vogtmanns outer space, and spaces of JSJ decompositions. We discuss what features are common to trees in a given deformation space, how to pass from one tree to all other trees in its deformation space, and the topology of deformation spaces. In particular, we prove that all deformation spaces are contractible complexes.