Titles, abstracts and notes
Title: Translation lengths in the free factor complex (LECTURE NOTES)
Abstract: This is work in progress, joint with Camille Horbez and RicWade. It is difficult to give explicit lower bounds to the distances in the free factor complex FF_N. We tackle the conjecture that there is a uniform lower bound to the translation length of an iwip in FF_N and we prove it under some extra conditions on the train track representative. The main tool is the geometry of folding paths.
Title: Untwisted outer space and the Torelli subgroup for right-angled Artin groups (LECTURE NOTES)
Abstract: Let G be a right-angled Artin group with outer automorphism group Out(G). Charney-Stambaugh-Vogtmann introduced a finite-dimensional, contractible simplicial complex K(G) on which a certain subgroup of Out(G) acts properly, discontinuously and cocompactly. We review the construction of the untwisted outer space K(G) and use it to give a geometric proof that the Torelli subgroup of Out(G) is torsion-free.
Title: Tree substitution for parageometric iwips. (LECTURE NOTES)
Abstract: (Joint with Milton Minervino) The repelling tree of an iwip automorphism has a self-similar structure which is described by a graph-directed iterated functions system. In the context of substitutions (that-is-to-say positive automorphisms) this is a long studied fractal construction through prefix-suffix automaton and Rauzy fractal. We detail how the singular leaves of the attracting lamination can be identified with gluing points in the repelling tree. We provide an algorithm to draw this tree using a tree substitution. As a by-product of this construction we get a contour substitution which is a pseudo-Anosov hanging over any parageometric iwip.
Abstract: Consider the class of outer automorphism groups of RAAGs (OARs). The restriction map technique of Charney--Crisp--Vogtmann is a powerful tool for studying OARs, because it leads to a template for inductive arguments. However, the image of a restriction map on an OAR is hard to describe and is usually not an OAR; this limits the results that can be proven using this approach. In joint work with Ric Wade, we study restriction maps on relative outer automorphism groups of RAAGs (ROARs), which are the conjugacy stabilizers in OARs of certain collections of subgroups. In particular, every OAR is a ROAR. We give a precise description of the image of a restriction map on a ROAR: it is always a simpler ROAR. Further, the kernel of such a map is also a simpler ROAR. This gives us a stronger inductive template for a wider class of groups. As applications of this technique, we prove that ROARs (and OARs) are of type VF, and we prove a general structural result for ROARs.
Title: On ranks of hyperbolic group extensions. (LECTURE NOTES)
Abstract: The rank of a group is the minimal cardinality of a generating set. While simple to define, this quantity is notoriously difficult to calculate, and often uncomputable, even for well behaved groups. In this talk I will explain general conditions that may be used to show many hyperbolic group extensions have rank equal to the rank of the kernel plus the rank of the quotient and, further, that any minimal generating set is Nielsen equivalent to one in a standard form. This builds on work of Souto on fundamental groups of fibered hyperbolic three manifolds and of Scott-Swarup who in this setting proved that infinite-index finitely generated subgroups of the fiber are quasi-convex in the ambient group. As an application, we prove that if g_1,...,g_k are independent, atoroidal, fully irreducible outer automorphisms of the free group F_n, then there is a power m so that the subgroup generated by f_1^m,...,f_k^m gives rise to a hyperbolic extension of F_n of rank n+k. Joint work with Sam Taylor.
Title: The conjugacy problem for polynomially growing outer automorphisms of free groups (LECTURE NOTES)
Abstract: We discuss a solution to the conjugacy problem for (rotationless) polynomially growing outer automorphisms of F_n. The linear case is due to Cohen-Lustig and Krstic-Lustig-Vogtmann. This is joint work with Michael Handel.
Title: The boundary of relative free fact or graphs, and subgroup classificationfor automorphisms of free products. (LECTURE NOTES)
I will give a description of the Gromov boundary of the graph of free factors of a free product. This description allows to apply Horbez's random walk argument and prove a classification for subgroups of automorphisms of free products: either such a subgroup contains a relatively fully irreducible automorphism, or it virtually preserves a larger free factor system. This generalizes a result by Handel and Mosher. The Gromov boundary of the graph of cyclic factors will also be considered, with a corollary saying that if a subgroup of automorphisms doesn't virtually preserve relative a free factor system or a non-peripheral conjugacy class, then it contains an atoroidal fully irreducible automorphism. This is a joint work with Camille Horbez.
Title: Asymptotic dimension of geometric graphs (LECTURE NOTES)
Abstract: The curve graph of a surface of finite type is known to havefinite asymptotic dimension. W e discuss the asymptotic dimension of various substitutes forthe curve graph related to the handlebody group and the outer automorphismgroup of the free group. Some of these graphs have finite asymptoticdimension, others don't.If time permits, we will also discuss some applications and open questions.
Title: Out(F_n) and the handlebody group (LECTURE NOTES)
Abstract: The handlebody group is the group of isotopy classes of homeomorphisms of a three-dimensional handlebody. By restriction to theboundary it in j ects into the usual surface mapping class group, and by considering the action on the fundamental group it surjects to Out(F_n).In this talk we will discuss results which explore these connections and show that geometrically, it behaves like Out(F_n) rather than mapping class groups. Part of this is joint work with Ursula Hamenstädt.
Title: Word complexity of the attracting fixed points of a free group automorphism. (LECTURE NOTES)
Abstract: An automorphism of a free group F fixes some points in the Gromov boundary of F. Given a basis A of F, such a fixed point u is represented by a right infinite word. The word complexity p(n) of u counts the number of subwords of length n occuring in u, for all integer n. When u is an attracting fixed point, we show that there are 5 possibilities for the (class of the) function p(n) ( constant, linear, quadratic, n.log(n) and n.log(log(n)) ). The same result is true for the word complexity of the attracting laminations of an outer automorphism of F. This is joint work with Gilbert Levitt.
Title: Growth under automorphisms of hyperbolic groups (LECTURE NOTES)
Abstract: Let G be a torsion-free Gromov hyperbolic group, let S be a finite generating set of G, and let f be an automorphism of G. We investigate the possible growth types of the word length of f^n(g), where g is an element of G. Growth was completely described by Thurston when G is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel’s work on train-tracks when G is a free group. We address the case of a general torsion-free hyperbolic group, and show in particular that every element g has a well-defined exponential growth rate under iteration of f, and that only finitely many exponential growth rates arise as g varies in G. This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.
Title: The Thurston norm for automorphisms of free groups (LECTURE NOTES)
Abstract: We will discuss how methods stemming from the theory of L2-invariants can be used in adapting the definition of the Thurston norm to thesetting of free-by-cyclic groups. We will see that the unit ball of the resulting se mi-norm is a polytope which controls the ways in which the free-by-cyclic group can fibre (in complete analogy to the 3-manifold setting).
Title: Monodromies of free-by-cyclic groups. (LECTURE NOTES)
Abstract: Given an outer automorphism of a free group, consider the associated free-by-cyclic group. This can often be expressed as a free-by-cyclic group in many different ways, giving rise to a families of outer automorphisms of free groups. I'll discuss work with Dowdall and Kapovich on relationships among the automorphisms in this family and tools for studying them.
Title: On automorphisms leaving a random subgroup invariant. (LECTURE NOTES)
Abstract: I'll discuss the validity of the slogan saying that a random subgroup is left invariant by very few automorphisms (joint with Vincent Guirardel).
Title: On automorphisms groups of RAAGS (LECTURE NOTES)
Abstract: We consider several problems on automorphisms groups of RAAGs. First, we consider the case when the defining graph is a tree and study from a probabilistic point of view the value of the first Betti number of certain finite subgroup.
Then, we look at a different subgroup of the full automorphism group: the subgroup generated by partial conjugations. We construct a normal subgroup of this group which is a RAAG itself and study the problem of when the Lie algebra of the lower central series of this group is Koszul. This is a joint work with Javier Aramayona, José Fernández, Pablo Fernández aand Luis Mendoça.
Title: Loops with Large Twist Get Short Along Quasi-geodesics in Out(F_n) (LECTURE NOTES)
Abstract: We study the behaviour of quasi-geodesics in Out(F_n). Given an element \phi in Out(F_n) there are several natural paths connecting the origin to \phi in Out(F_n), for example, a path associate to sequence of Stalling folds and paths induced by the shadow of standard geodesics in Outer space. We show that neither of these paths is, in general, a quasi-geodesic in Out(F_n). In fact, we construct examples where any quasi-geodesic connecting \phi to the origin will have to back-track in some free factor of F_n.
Title: The outer automorphism group of a right-angled Coxeter group is either large or virtually abelian (LECTURE NOTES)
Abstract: In the study of automorphisms of graph products of cyclic groups (including RAAGs and RACGs), a separating intersection of links (SIL) has been shown to hold a lot of power. The reason for this is that a SIL is exactly the necessary condition on the underlying graph that determines when two partial conjugations do not commute. We introduce two variations on a SIL that give a combinatorial condition on a right-angled Coxeter group that determine the dichotomy given in the title. This is joint work with Tim Susse.
Title: The abstract commensurator of Out(F_3) (LECTURE NOTES)
Abstract: A theorem of Farb and Handel states that when n is greater than or e qualto 4, every isomorphism between two finite index subgroups of Out(F_n)is induced by conjugation in the group. In joint work with Martin Bridson and Camille Horbez, we show that this is also true in the case when n=3. The proof proceeds in the spirit of Ivanov's work on the mapping class group and utilizes the action of Out(F_3) and its subgroups on relative free factor graphs and their boundaries. Time permitting, I will also discuss generalizations of the proof to other normal subgroups of Out(F_3) or in the case where n is arbitrary.
Title: Negative immersions for one-relator groups (LECTURE NOTES)
Abstract: The *primitivity rank* of word w in a free group, \pi(w), is the smallest rank of a subgroup of the free group F that contains w as an imprimitive element. I’ll explain recent results which show that \pi(w) has a profound influence over the subgroup structure of the one-relator group F/<<w>>. In particular, every subgroup generated by fewer than \pi(w) elements is free.
This joint work with Larsen Louder.